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samples/texts/1340369/page_1.md

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+Weird integrals
+
+$$ \pi \cot(\pi x) = \frac{1}{x} - 2 \int_{0}^{\infty} \frac{\sinh(xt)}{e^t - 1} dt \quad \text{for } |x| < 1 $$
+
+$$ \pi^2 \csc^2(\pi x) = \frac{1}{x^2} + 2 \int_0^\infty \frac{t \cosh(xt)}{e^t - 1} dt \quad \text{for } |x| < 1 $$
+
+Obviously the second one is the result of differentiating and negating the first.
+
+Tested both numerically. Both work well as long as $|x|$ isn't too close to 1.

samples/texts/4807413/page_10.md

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+yield another Cantor set, the only remaining case to consider
+is $X = \mu_0 \times \prod_{i=1}^k X_i$ where each $X_i$ is homeomorphic to $\mu_n$ for
+some $n \ge 1$. The path components of $X$ are of the form $\{p\} \times
+\prod_{i=1}^k X_i$ where $p \in \mu_0$. Since self homeomorphisms of $X$ can not
+take two points in a single path component to points in distinct
+path components, it follows that $X$ is not 2-homogeneous. $\square$
+
+The preceding corollary provides a partial negative answer
+to the question in [KKT] as to whether any finite or countable
+product of $\mu_1$ with itself is 2-homogeneous. Also, $\mu_n \times \mu_n$
+provides an example of a homogeneous space that is (2n)-dimensional, that is $LC^{n-1}$, and that is not 2-homogeneous.
+We now proceed to produce an example of a homogeneous $LC^{n-1}$space of dimension ($n+2$) that is not 2-homogeneous.
+
+**Theorem 4.4.** Let $X = \mu_n \times Y$ where $n \ge 1$, $Y$ is compact, path connected, homogenous and locally homologically Čech $n$-connected. Then $X$ is not 2-homogeneous.
+
+*Proof:* We proceed as in the proof of Theorem 4.1. Fix distinct points $x$ and $y$ in $\mu_n$. Choose distinct points $r$ and $s$ in $Y$. We will show that there is no homeomorphism $h : X \to X$ such that $h(\{(x,r), (x,s)\}) = \{(x,r), (y,s)\})$. Assume to the contrary that there is such a homeomorphism $h$.
+
+Let $p$ be projection of $X$ onto $\mu_n$,and let $q$ be projection
+of $X$ onto $Y$. Choose a neighborhood $U$ of $x$ in $\mu_n$ such that
+$p(h(U \times \{r\})) \cap p(h(U \times \{s\})) = \emptyset$, such that $q(h(U \times \{r\})) \cap
+q(h(U \times \{s\})) = \emptyset$, and such that the inclusion induced ho-
+momorphisms from the $n$-th Čech homology of $q(h(U \times \{r\}))$
+and from the $n$-th Čech homology of $q(h(U \times \{s\}))$ to the $n$-th
+Čech homology of $Y$ are trivial.
+
+As in Theorem 4.1, choose an embedding $e: S^n \to U$ and
+let $f_1: S^n \to X$ be the map given by $f_1(a) = (e(a), \{r\})$ and
+let $f_2: S^n \to X$ be the map given by $f_2(a) = (e(a), \{s\})$.
+Since $Y$ is path connected, the maps $f_1$ and $f_2$ are homotopic.
+As before, the map $e$ is essential with respect to $n$-th Čech
+homology. It follows that the maps $f_1$ and $f_2$ are also essential

samples/texts/4807413/page_11.md

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+with respect to *n*-th Čech homology, and so the maps $h \circ f_1$ and $h \circ f_2$ are essential with respect to *n*-th Čech homology.
+
+Since $p \circ h \circ f_1$ and $p \circ h \circ f_2$ are homotopic and have disjoint images, Lemma 3.2 again implies that these maps are not essential with respect to *n*-th Čech homology.
+
+So both $q \circ h \circ f_1$ and $q \circ h \circ f_2$ are essential with respect to *n*-th Čech homology. But this contradicts the fact that the homeomorphisms from the *n*-th Čech homology of $q(h(U \times \{r\})$ and from the *n*-th Čech homology of $q(h(U \times \{s\}))$ to the *n*-th Čech homology of $Y$ are trivial. So there is no homeomorphism $h$ as assumed, and it follows that $X$ is not 2-homogeneous.
+
+**Corollary 4.5.** The product of $\mu_n$, $n \ge 1$ with any ANR or with any manifold is not 2-homogeneous.
+
+**Corollary 4.6.** For each positive integer $n$, there is an $(n+1)$-dimensional homogeneous compact metric space that is $(n-1)$-connected and is not 2-homogeneous.
+
+*Proof:* That the space $\mu_n \times S^1$ satisfies the conditions in the corollary follows directly from the previous theorem. $\square$
+
+Note that the techniques used in proving Theorems 4.2 and 4.4 can be used to prove that any finite product of Menger spaces and manifolds, or any finite product of Menger spaces and spaces that are locally homologically Čech $n$-connected for sufficiently many values of $n$, are not 2-homogeneous. These techniques can also be used to prove that such products are not $n$-homogeneous for values of $n$ greater than 2. One can also use the techniques in the above theorems to analyze the type of self homeomorphisms of finite products of Menger spaces as was done in [KKT] for the product $\mu_1 \times \mu_1$.
+
+## 5. QUESTIONS
+
+The following questions remain open.

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+**Question 1.** Is there a compact metric space of dimension less than ($n + 2$) that is homogeneous, locally $n$-connected, and not 2-homogeneous?
+
+**Question 2.** If a homogeneous compact metric space is locally $n$-connected for all $n$, is the space necessarily 2-homogeneous?
+
+## REFERENCES
+
+[An1] R. D. Anderson, *A characterization of the universal curve and a proof of its homogeneity*, Ann. of Math., **68** (1958), 313-324.
+
+[An2] —, *1-dimensional continuous curves and a homogeneity theorem*, Ann. of Math., **68** (1958), 1-16.
+
+[Be] M. Bestvina, *Characterizing k-dimensional universal Menger compacta*, Memoirs Amer. Math. Soc., **71** (1988), no.380.
+
+[CF] M. L. Curtis and M. K. Fort, *The fundamental group of one-dimensional spaces*, Proc. Amer. Math. Soc., **10** (1959), 140-148.
+
+[ES] S. Eilenberg and N. Steenrod, *Foundations of Algebraic Topology*, Princeton University Press, Princeton, New Jersey, 1952.
+
+[Ga] D. J. Garity, *On multiple homogeneity of products of Menger spaces*, Proc. of the Ninth Annual Workshop in Geometric Topology (J. Henderson, F. Tinsley ed.), Colorado College, Colorado Springs, Colorado, 1992, 18-24.
+
+[Kel] J. Kennedy Phelps, *Homomorphisms of products of universal curves*, Houston J. Math., **6** (1980), 127-143.
+
+[Ke2] —, *A condition under which 2-homogeneity and representability are the same in continua*, Fund. Math., **121** (1984), 89-98.
+
+[Ku] K. Kuperberg, *On the bihomogeneity problem of Knaster*, Trans. Amer. Math. Soc., **321** (1990), 129-143.
+
+[KKT] K. Kuperberg, W. Kuperberg, and W. R. R. Transue, *On the 2-homogeneity of Cartesian products*, Fund. Math. CX (1980), 131-134.
+
+[KKT2] —, *Homology separation and 2-homogeneity*, Continua with the Houston problem book (H. Cook, W. T. Ingram, K. T. Kuperberg, A. Lelek, P. Minc, ed.), vol. 170, Marcel Decker, New Your, 1995.
+
+[Mg] K. Menger, *Kurventheorie*, Teubner, Berlin - Leipzig, 1932.
+
+[Mu] J. Munkres, *Elements of Algebraic Topology*, Benjamin/Cummings, Menlo Park, California, 1984.
+
+[Un] G. S. Ungar, *On all kinds of homogeneous spaces*, Trans. Amer. Math. Soc., **212** (1975), 393-400.

samples/texts/4807413/page_13.md

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+Oregon State University
+Corvallis, OR 97331
+e-mail: garity@math.orst.edu

samples/texts/4807413/page_8.md

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+phism $\lambda$ to $\theta$ can be constructed that takes elements of the
+form $[\alpha_j \otimes \beta_{n-j}]$ to $[\alpha_j] \otimes [\beta_{n-j}]$ in $H_j(M_n^i) \otimes H_{n-j}(X_i)$ when
+$\alpha_j$ and $\beta_{n-j}$ are cycles, and takes $[\alpha_j \otimes \beta_{n-j}]$ to 0 otherwise.
+Since $H_j(M_n^i)$ is nonzero only for $j=0$ or $j=k$, it follows
+that
+
+$$
+\lambda \circ u([c]) = [p_{i\#}(c)] \otimes [q_{i\#}(cB_0)] + [p_{i\#}(cF_0)] \otimes [q_{i\#}(c)]
+$$
+
+Since $\lambda$ and $u$ are isomorphisms, if $[p_{i\#}(c)]$ and $[q_{i\#}(c)]$ are both 0, then $[c] = 0$.
+
+Let $\alpha = (\alpha_1, \alpha_2, \ldots, \alpha_n, \ldots)$ be a nontrivial element of the k-th Čech homology of Y that is given by the hypotheses. That is, each $\alpha_i = f_{i*}(\gamma)$ where $\gamma$ is a generator of $H_k(S^k)$ and where $f_i$ is the composition of $f$ with inclusion from Y to $Y_i$. Then there is a positive integer $k$ such that for all $i \ge k$, $\alpha_i \ne 0$ in $H_k(Y_i)$. By the previous paragraph, either $p_{i*}(\alpha_i)$ or $q_{i*}(\alpha_i)$ is nontrivial for all $i \ge k$. The result now follows by considering the definition of the induced homomorphism on Čech homology groups.
+
+If in fact $k < n$, then each $p_{i*}(\alpha_i)$ is trivial, so that $p_*(\alpha)$ is trivial, and thus $q \circ f$ is essential with respect to $k$-th Čech homology. $\square$
+
+4. THE MAIN RESULTS
+
+**Definition 4.1.** A space X is locally homologically Čech *n*-connected if for each point *p* ∈ X and for each neighborhood U of *p*, there is a neighborhood V of *p* such that the inclusion induced homomorphism from the *n*-th Čech homology of V to the *n*-th Čech homology of U is trivial.
+
+**Theorem 4.2.** Let $X = \mu_n \times \prod_{i=1}^k Y_i$ where $n \ge 1$, and where each $Y_i$ is homeomorphic to $\mu_j$ for some $j \ge n$. Then $X$ is not 2-homogeneous.
+
+*Proof:* Fix distinct points x and y in μₙ. Choose points r and s in Πk[i=1 Yᵢ so that for each i, 1 ≤ i ≤ k, qᵢ ≠ rᵢ. We will show that there is no homeomorphism h : X → X such that

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+$h(\{(x, r), (x, s)\} = \{(x, r), (y, s)\}). Assume to the contrary that there is such a homeomorphism $h$.
+
+Let $p$ be projection of $X$ onto $\mu_n$, let $q$ be projection of $X$ onto $\prod_{i=1}^k Y_i$ and let $q_i$ be projection of $X$ onto $Y_i$. Choose a neighborhood $U$ of $x$ in $\mu_n$ such that $p(h(U \times \{r\}) \cap p(h(U \times \{s\})) = \emptyset$, and such that for each $i$, $q_i(h(U \times \{r\}) \cap q_i(h(U \times \{s\})) = \emptyset$. Choose an embedding $e: S^n \to U$. This is possible by Lemma 3.2. Let $f_1: S^n \to X$ be the map given by $f_1(a) = (e(a), \{r\})$ and let $f_2: S^n \to X$ be the map given by $f_2(a) = (e(a), \{s\})$. Since $\prod_{i=1}^k X_i$ is path connected, the maps $f_1$ and $f_2$ are homotopic. By Lemma 3.1, the map $e$ is essential with respect to $n$-th Čech homology. It follows that the maps $f_1$ and $f_2$ are also essential with respect to $n$-th Čech homology, and so the maps $h \circ f_1$ and $h \circ f_2$ are essential with respect to $n$-th Čech homology.
+
+Lemma 3.3 implies that for each $i$, either $p \circ h \circ f_i$ or $q \circ h \circ f_i$ is essential with respect to $n$-th Čech homology. Since $f_1$ and $f_2$ are homotopic, this implies that either both $p \circ h \circ f_1$ and $p \circ h \circ f_2$ are essential with respect to $n$-th Čech homology, or both $q \circ h \circ f_1$ and $q \circ h \circ f_2$ are essential with respect to $n$-th Čech homology. Since $p \circ h \circ f_1$ and $p \circ h \circ f_2$ are homotopic and have disjoint images, Lemma 3.2 implies that these maps are not essential with respect to $n$-th Čech homology.
+
+So both $q \circ h \circ f_1$ and $q \circ h \circ f_2$ are essential with respect to $n$-th Čech homology. Repeated application of Lemma 3.2 shows that for each $i$, both $q_i \circ h \circ f_1$ and $q_i \circ h \circ f_2$ are not essential with respect to $n$-th Čech homology. This leads to a contradiction. So there is no homeomorphism $h$ as assumed, and it follows that $X$ is not 2-homogeneous. $\square$
+
+**Corollary 4.3.** Any finite product of two or more Menger spaces, where at least one of the Menger spaces is not $\mu_0$, is not 2-homogeneous. In particular, $\mu_m \times \mu_n$, where $\max\{m, n\} \ge 1$ is not 2-homogeneous.
+
+*Proof:* The case where each factor is not $\mu_0$ follows directly from the previous theorem. Since finite products of Cantor sets

samples/texts/486326/page_1.md

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+7th October
+
+Foundation Plus 5-a-day
+
+Work out
+
+$$3\frac{2}{3} - 2\frac{3}{4}$$
+
+A spinner has two sections, yellow and blue.
+
+The probability of the spinner landing on yellow is $\frac{3}{5}$
+
+1st Spin
+
+2nd Spin
+
+Yellow
+$\frac{3}{5}$
+Blue
+Yellow
+Blue
+
+The spinner is spun twice.
+
+Find the probability that the spinner lands on blue twice.
+
+Find the probability that the spinner lands on different colours.
+
+Expand and simplify
+
+$$(x - 10)(x + 8)$$
+
+Rachel buys a DVD for £18.50.
+
+A year later she sells it for £15.91
+
+What is the percentage decrease in value of the DVD?

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+# Partial angular coherence and the angular Schmidt spectrum of entangled two-photon fields
+
+Anand Kumar Jha,¹ Girish S. Agarwal,² and Robert W. Boyd¹
+
+¹The Institute of Optics, University of Rochester, Rochester, New York 14627, USA
+
+²Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA
+
+(Received 13 September 2011; published 27 December 2011)
+
+We study partially coherent fields that have a coherent-mode representation in the orbital-angular-momentum-mode basis. For such fields, we introduce the concepts of the angular coherence function and the coherence angle. Such fields are naturally produced by the process of parametric down-conversion—a second-order nonlinear optical process in which a pump photon breaks up into two entangled photons, known as the signal and idler photons. We show that the angular coherence functions of the signal and idler fields are directly related to the angular Schmidt (spiral) spectrum of the down-converted two-photon field and thus that the angular Schmidt spectrum can be measured directly by measuring the angular coherence function of either the signal or the idler field, without requiring coincidence detection.
+
+DOI: 10.1103/PhysRevA.84.063847
+
+PACS number(s): 42.50.Ar, 42.65.Lm, 03.65.Ud, 42.25.Kb
+
+## I. INTRODUCTION
+
+Classical coherence theory is a well-established subject. Its modern interpretation is largely due to Wolf and co-workers [1,2]. The study of coherence in the context of entangled fields has revealed a much deeper understanding of entanglement itself [3–6]. One of the central concepts of classical coherence theory is the cross-spectral-density function, which quantifies the field correlations in the space-frequency domain. A cross-spectral-density function always has a unique coherent-mode representation, which is a way to represent a partially coherent field as an incoherent sum of a finite number of completely coherent fields. In this paper, we study the partially coherent fields that have a coherent-mode representation in the orbital-angular-momentum basis. We show that, for such fields, it is very useful to introduce the concepts of the angular coherence function and the coherence angle. In fact, such fields are naturally produced by the process of parametric down-conversion (PDC), owing to the conservation of orbital angular momentum (OAM) in parametric down-conversion [7,8].
+
+The OAM entanglement of PDC photons [8] is a great resource for quantum-information-based protocols due to the fact that the OAM basis provides a discrete but infinite-dimensional Hilbert space, as opposed to the polarization basis, which provides only a two-dimensional Hilbert space [9,10]. For this reason, an accurate measurement of the dimensionality of the OAM-entangled photons is very important. There are two generic ways in which the dimensionality can be measured. The first is by directly measuring the two-photon intensity in coincidence at different values of the OAM mode indices [11,12] and the second is by using the Hong-Ou-Mandel interference technique [13]. In this paper, we propose an alternative way [14] of measuring the dimensionality, by measuring the angular coherence function of either of the down-converted photons. This scheme is different from the existing schemes in that it requires only singles detection, as opposed to the coincidence detection required in the other schemes.
+
+## II. PARTIALLY COHERENT FIELDS
+
+### A. General representation
+
+In this section, we briefly describe the general representation of partially coherent fields. Let {$V(\mathbf{r},t)$} be an
+
+ensemble representing the statistical properties of a partially coherent field that is both stationary, at least in the wide sense, and ergodic. One way to characterize the statistical correlations of such fields is through the mutual coherence function $\Gamma(\mathbf{r}_1,\mathbf{r}_2,\tau)$, which quantifies the field correlation between the space-time points $(\mathbf{r}_1,t)$ and $(\mathbf{r}_2,t+\tau)$, and is defined as $\Gamma(\mathbf{r}_1,\mathbf{r}_2,\tau) = \langle V^*(\mathbf{r}_1,t)V(\mathbf{r}_2,t+\tau) \rangle$, where $\langle \cdots \rangle_e$ represents the ensemble average. Another way, which is more convenient, to characterize the field correlations is through the cross-spectral-density function $W(\mathbf{r}_1,\mathbf{r}_2,\omega)$, which quantifies the field correlations in the space-frequency domain and is defined as
+
+$$W(\mathbf{r}_1, \mathbf{r}_2, \omega) \equiv \frac{1}{2\pi} \int_{-\infty}^{\infty} \Gamma(\mathbf{r}_1, \mathbf{r}_2, \tau) e^{i\omega\tau} d\tau. \quad (1)$$
+
+For conceptual clarity, we suppress from now on the frequency argument in the definition of the cross-spectral-density function. We also assume that the cross-spectral-density function is a continuous function of $\mathbf{r}_1$ and $\mathbf{r}_2$ within the domain $D$ of interest. The cross-spectral-density function is a bounded function, in the sense that
+
+$$\int_D \int_D |W(\mathbf{r}_1, \mathbf{r}_2)|^2 d\mathbf{r}_1 d\mathbf{r}_2 < \infty. \quad (2)$$
+
+Further, it is a Hermitian function, that is,
+
+$$W^*(\mathbf{r}_1, \mathbf{r}_2) = W(\mathbf{r}_2, \mathbf{r}_1). \quad (3)$$
+
+And most importantly, it is a non-negative definite function, that is,
+
+$$\int_D \int_D W(\mathbf{r}_1, \mathbf{r}_2) f^*(\mathbf{r}_1) f(\mathbf{r}_2) d\mathbf{r}_1 d\mathbf{r}_2 \ge 0, \quad (4)$$
+
+where $f(\mathbf{r})$ is any square integrable function. The physical interpretation of the non-negative-definiteness condition is that the intensity distribution produced by the field, with an aperture function $f(\mathbf{r})$ in domain $D$, on a screen is always non-negative. The above conditions, along with the multidimensional version of the Mercer theorem, imply that the cross-spectral-density function $W(\mathbf{r}_1,\mathbf{r}_2)$ is a Hilbert-Schmidt kernel and that it has a coherent-mode representation of the form [2]
+
+$$W(\mathbf{r}_1, \mathbf{r}_2) = \sum_n \alpha_n \psi_n^*(\mathbf{r}_1) \psi_n(\mathbf{r}_2). \quad (5)$$

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+The functions $\psi_n^*(\mathbf{r})$ are the eigenfunctions and the coefficients $\alpha_n$ are the eigenvalues of the integral equation $\int W(\mathbf{r}_1, \mathbf{r}_2)\psi_n(\mathbf{r}_1)d\mathbf{r}_1 = \alpha_n\psi_n(\mathbf{r}_2)$. The Hermiticity and the non-negative definiteness of $W(\mathbf{r}_1, \mathbf{r}_2)$ ensure that the integral equation has at least one nonzero eigenvalue and that all the eigenvalues are real and non-negative, i.e., $\alpha_n \ge 0$. The above equation can be rewritten as $W(\mathbf{r}_1, \mathbf{r}_2) = \sum_n \alpha_n W^{(n)}(\mathbf{r}_1, \mathbf{r}_2)$, where $W^{(n)}(\mathbf{r}_1, \mathbf{r}_2) \equiv \psi_n^*(\mathbf{r}_1)\psi_n(\mathbf{r}_2)$. This representation implies that for any partially coherent field there exists at least one basis in which the cross-spectral-density function can be represented as a superposition of modes that are completely coherent in the space-frequency domain.
+
+**B. Partially coherent field in the Laguerre-Gaussian basis**
+
+Every type of partially coherent field is characterized by its unique coherent-mode representation. In this paper, we are investigating partially coherent fields that have a coherent-mode representation in the Laguerre-Gaussian (LG) basis. A coherent mode in the LG basis is referred to as an LG mode or an LG beam; these are the exact solutions of the paraxial Helmholtz equation. The normalized field amplitude of these modes at $z=0$ in the cylindrical coordinate system is given by
+
+$$
+\begin{align}
+[\mathrm{LG}_p^l(\rho, \phi)] &\equiv [\mathrm{LG}_p^l(\rho)]e^{il\phi} = \sqrt{\frac{2p!}{\pi(|l|+p)!}} \nonumber \\
+&\quad \times \frac{1}{w_0}\left(\frac{\sqrt{2}\rho}{w_0}\right)^{|l|} L_p^l\left(\frac{2\rho^2}{w_0^2}\right) \exp\left(-\frac{\rho^2}{w_0^2}\right)e^{il\phi}, \tag{6}
+\end{align}
+$$
+
+where $w_0$ is the beam waist radius at $z=0$ and $l$ is the azimuthal mode index. Due to the azimuthal phase dependence of $e^{il\phi}$, these modes carry an orbital angular momentum of $lẖ$ per photon [15]. These modes have been extensively studied in the last few decades. Such fields are very important as they hold promise for many new fascinating applications, especially in quantum-information science. The type of partially coherent fields that we consider in this paper has the following coherent-mode representation:
+
+$$
+\begin{align}
+W(\mathbf{r}_1, \mathbf{r}_2) &\rightarrow W(\rho_1, \phi_1; \rho_2, \phi_2) \nonumber \\
+&= \langle V^*(\rho_1, \phi_1)V(\rho_2, \phi_2) \rangle_e \nonumber \\
+&= \sum_{l,p,p'} \alpha_{lp'} [\mathrm{LG}_p^{*l}(\rho_1, \phi_1)][\mathrm{LG}_{p'}^{l}(\rho_2, \phi_2)], \tag{7}
+\end{align}
+$$
+
+where $V(\rho,\phi)$ is a single realization of the field at location $(\rho,\phi)$. We note that the field is a coherent superposition of modes carrying different values of the orbital-angular-momentum mode index $l$. The specific question that we now ask is the following: “For a field that has the above form for the cross-spectral-density function, what is the correlation between the fields at two different angular positions, after the correlations have been integrated over the radial dimensions?” In order to answer this question, we consider the situation as shown in Fig. 1. Consider a partially coherent field passing through a screen in the form of a double angular slit. The two slits are centered at angular positions $\phi_1$ and $\phi_2$, respectively.
+
+FIG. 1. (Color online) A scheme for studying the angular coherence properties of a partially coherent beam.
+
+The separation between the slits is $\Delta\phi = \phi_1 - \phi_2$. The field $\Psi(\rho,\phi)$ immediately after the aperture is given by
+
+$$
+\Psi(\rho, \phi) = V(\rho, \phi)\Phi(\phi), \quad (8)
+$$
+
+where $V(\rho,\phi)$ is the incoming field and $\Phi(\phi)$ is the amplitude transmission function of the aperture. We decompose the above field in the LG basis as
+
+$$
+\Psi(\rho, \phi) = \sum_{l,p} A_{lp} [\mathrm{LG}_p^l(\rho, \phi)], \quad (9)
+$$
+
+where
+
+$$
+A_{lp} = \iint \rho d\rho d\phi [\mathrm{LG}_p^{kl}(\rho, \phi)] V(\rho, \phi) \Phi(\phi) \quad (10)
+$$
+
+is the probability amplitude for the field to be found in mode $\mathrm{LG}_p^l(\rho, \phi)$. Since we are interested in field correlations at different angular positions, we sum over all the $p$ modes and obtain the intensity $I_l$ of the field for a given value of $l$ as
+
+$$
+I_l = \sum_{p=0}^{\infty} I_{lp} = \sum_{p=0}^{\infty} \langle A_{lp}^* A_{lp} \rangle_e . \qquad (11)
+$$
+
+Using Eqs. (7) and (10), we write $I_l$ as
+
+$$
+\begin{equation}
+\begin{split}
+I_l = {}& \sum_{l', p', p''} \alpha_{l'p'p''} \sum_p \iint \rho \rho' d\rho d\rho' [\mathrm{LG}_p^l(\rho)] \\
+         & \times [\mathrm{LG}_p^{*l}(\rho')] [\mathrm{LG}_{p'}^{*l'}(\rho)] [\mathrm{LG}_{p''}^{l''}(\rho')] \\
+         & \times \iint d\phi d\phi' e^{i(l-l')(\phi-\phi')} \Phi^*(\phi) \Phi(\phi'),
+\end{split}
+\tag{12}
+\end{equation}
+$$
+
+where we have substituted for $\langle V^*(\rho_1, \phi_1)V(\rho_2, \phi_2) \rangle_e$ from Eq. (7). The summation over $p$ can be evaluated by using the identity $\sum_p [\mathrm{LG}_p^l(\rho)][\mathrm{LG}_{p'}^{*l'}(\rho')] = (1/\pi)\delta(\rho^2 - \rho'^2)$ (see Appendix B for the derivation), which gives
+
+$$
+I_l = \sum_{l', p', p''} \alpha_{l'p'p''} \frac{1}{2\pi} \int d\rho d\rho' [\mathrm{LG}_{p'}^{*l'}(\rho)][\mathrm{LG}_{p''}^{l''}(\rho)] \\
+\times \iint d\phi d\phi' e^{i(l-l')(\phi-\phi')} \Phi^*(\phi)\Phi(\phi'). \quad (13)
+$$
+
+The radial integral is evaluated by noting that the radial LG modes with a fixed value for the angular-momentum-mode index form a complete basis, that is, $\int \rho d\rho [\mathrm{LG}_{p'}^{*l'}(\rho)][\mathrm{LG}_{p''}^{l''}(\rho)] = \delta_{p'p''}/2\pi$. Using this formula, we obtain
+
+$$
+I_l = \sum_{l'} \frac{C_{l'}}{2\pi} \iint d\phi d\phi' e^{i(l-l')(\phi-\phi')} \Phi^*(\phi)\Phi(\phi'), \quad (14)
+$$

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+where $C_l' = (1/2\pi) \sum_{p'} \alpha_{l'p'p'}$. Next, we substitute the expression for the aperture function $\Phi(\phi) = k_1\delta(\phi - \phi_1) + k_2\delta(\phi - \phi_2)$. The intensity $I_l$ then assumes the following form:
+
+$$ I_l = \frac{k_1^2}{2\pi} \sum_{l'=-\infty}^{\infty} C_{l'} + \frac{k_2^2}{2\pi} \sum_{l'=-\infty}^{\infty} C_{l'} + \frac{k_1 k_2}{2\pi} \sum_{l'=-\infty}^{\infty} C_{l'} e^{-il'\Delta\phi} e^{il'\Delta\phi} + \text{c.c.,} \quad (15) $$
+
+Equation (15) can be seen to be the angular interference law, since it quantifies the interference between the fields coming from two separate angular positions.
+
+The function
+
+$$ W(\phi_1, \phi_2) = \sum_{l'=-\infty}^{\infty} C_{l'} e^{-il'\Delta\phi} \quad (16) $$
+
+represents the correlation that exists between the fields at $\phi_1$ and $\phi_2$. We refer to $W(\phi_1, \phi_2)$ as the angular coherence function. We note that in Ref. [16] Paterson introduced the “rotational coherence function,” which describes correlation between two field points with the same radial but different angular positions. The angular correlation function constructed above is integrated over the radial dimensions and thus describes only the correlation between field points with different angular positions, without any reference to their radial positions. The field represented by $W(\phi_1, \phi_2)$ is completely coherent if there is only one term in the above expansion. However, when there is more than one term in the expansion, the field is only partially coherent, and, as a consequence, two field points are mutually coherent over only a finite range of angular separation $\Delta\phi$. In order to quantify this thought, we rearrange the above equation to write it as
+
+$$ I_l = \frac{1}{2\pi} \sum_{l'=-\infty}^{\infty} C_{l'} [k_1^2 + k_2^2 + 2k_1 k_2 \lambda(\Delta\phi) \cos(l\Delta\phi + \theta)], \quad (17) $$
+
+where
+
+$$ \lambda(\Delta\phi) = \frac{|W(\phi_1, \phi_2)|}{\sum_{l'=-\infty}^{\infty} C_{l'}} \quad (18) $$
+
+is the degree of angular coherence and $\theta$ the argument of $W(\phi_1, \phi_2)$. For a completely coherent field, the degree of coherence $\lambda(\Delta\phi)$ is equal to unity. The width of $\lambda(\Delta\phi)$ is a measure of the angular separation over which the field remains coherent. We note that $\lambda(\Delta\phi)$ involves a discrete Fourier transform and that therefore it is a periodic function of the argument $\Delta\phi$. For this reason, one has to be careful in defining the width of $\lambda(\Delta\phi)$. However, when $C_{l'}$ has a broad distribution in $l'$ such that the spread of $\lambda(\Delta\phi)$ as a function of $\Delta\phi$ is well within the range $[0, 2\pi]$, the width of $\lambda(\Delta\phi)$ can be defined unambiguously, and this width can, to a very good approximation, be taken as the coherence angle of the field. As shown in Appendix A, when $C_{l'}$ has a broad Gaussian distribution in $l'$ with $\sigma$ being the standard deviation of the distribution, $\lambda(\Delta\phi)$ assumes, to within a very good approximation, the following functional form:
+
+$$ \lambda(\Delta\phi) = \exp\left(-\frac{\sigma^2 \Delta\phi^2}{2}\right). \quad (19) $$
+
+FIG. 2. (Color online) Intensity $I_l$, with $l = 30$ and $k_1 = k_2$, as a function of the angular separation $\Delta\phi$ for three different values of $\sigma$: (a) $\sigma = 1$, (b) $\sigma = 4$, and (c) $\sigma = 8$.
+
+We note that $1/\sigma$ is a measure of the angular width over which the fields at the two angular positions remain mutually coherent. Therefore, $\phi_{coh} \equiv 1/\sigma$ can be defined as the coherence angle of the beam. Figure 2 shows plots of the detection probability $I_l$ as a function of the angular separation $\Delta\phi$ for three different values of $\sigma$. We see that as the width $\sigma$ of the OAM-mode distribution increases, the coherence angle decreases. The visibility of angular interference is given by
+
+$$ V(\Delta\phi) = \frac{2k_1 k_2}{k_1^2 + k_2^2} \lambda(\Delta\phi), \quad (20) $$
+
+and when $|k_1| = |k_2|$, we get $V(\Delta\phi) = \lambda(\Delta\phi)$.
+
+### III. ANGULAR COHERENCE AND OAM ENTANGLEMENT
+
+In this section, we study a process known as parametric down-conversion that produces fields of the type considered in the previous section. We also show how the concept of angular

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+coherence can be useful for characterizing OAM entanglement of the PDC photons [8].
+
+A. Field produced by parametric down-conversion
+
+Parametric down-conversion is a nonlinear optical process in which a pump photon is broken up into two entangled photons known as the signal photon and the idler photon. When the pump field is of the form of a Gaussian beam, that is, an LG beam with $l = 0$ and $p = 0$, the state $|\psi_2\rangle$ of the down-converted two-photon field is given by [17–21]
+
+$$ |\psi_2\rangle = \sum_{l_s, p_s, p_i} \int d\omega_s \chi_{l_s p_s p_i}(\omega_s) \\ \times |l_s, p_s, \omega_s\rangle_s |-l_s, p_i, \omega_0 - \omega_s\rangle_i, \quad (21) $$
+
+where $\chi_{l_s p_s p_i}(\omega_s)$ is the probability amplitude that the signal and idler photons are in the LG modes characterized by indices $(l, p_s)$ and $(-l, p_i)$, respectively. We note that, due to the conservation of OAM in PDC, the signal and idler photons have equal but opposite OAMs. In writing the above state, we have assumed that the pump field is monochromatic with frequency $\omega_0$. We have also assumed perfect frequency phase matching such that $\omega_s + \omega_i = \omega_0$, where $\omega_s$ and $\omega_i$ denote the frequencies of the signal and idler photons. The density operator corresponding to the above two-photon state is $\hat{\rho}_2 = |\psi_2\rangle\langle\psi_2|$. The density operator $\hat{\rho}_s$ corresponding to the signal field can be calculated by taking a partial trace over the idler modes, which gives
+
+$$ \hat{\rho}_s = \text{tr}_i \hat{\rho}_2 = \sum_{l_s, p_s, p'_s} \int d\omega_s C_{l_s p_s p'_s}(\omega_s) |l_s, p_s, \omega_s\rangle_s \langle l_s, p'_s, \omega_s|, \quad (22) $$
+
+where
+
+$$ C_{l_s p_s p'_s}(\omega_s) = \sum_{p''_i} \chi_{l_s p_s p''_i}(\omega_s) \chi^*_{l'_s p'_s p''_i}(\omega_s). \quad (23) $$
+
+Next, by using Glauber's method [22], we calculate the classical correlation function $G_s(\rho_1, \phi_1; \rho_2, \phi_2; \tau)$ corresponding to the density matrix for the signal photon:
+
+$$ G_s(\rho_1, \phi_1; \rho_2, \phi_2; \tau) = \text{tr}[\rho_s \hat{E}^{(-)}(\rho_1, \phi_1, t) \hat{E}^{(+)}(\rho_2, \phi_2, t + \tau)], \quad (24) $$
+
+where
+
+$$ \hat{E}^{(-)}(\rho_1, \phi_1, t) = \sum_{l_1, p_1} \int d\omega_s \hat{s}_{l_1 p_1}^{\dagger}(\omega) [\mathrm{LG}_{p_1}^{*l_1}(\rho_1, \phi_1)] e^{i\omega t}, $$
+
+etc., and where $\hat{s}_{l_1 p_1}^{\dagger}(\omega)$ is the signal-photon annihilation operator for mode $[\mathrm{LG}_{p_1}^{*l_1}(\rho_1, \phi_1)]e^{i\omega t}$. Carrying out the above trace, we obtain
+
+$$ G_s(\rho_1, \phi_1; \rho_2, \phi_2; \tau) = \sum_{l_s, p_s, p'_s} \int d\omega_s C_{l_s p_s p'_s}(\omega_s) [\mathrm{LG}_{p'_s}^{*l'_s}(\rho_1, \phi_1)] \\ \times [\mathrm{LG}_{p'_s}^{l'_s}(\rho_2, \phi_2)] e^{-i\omega_s \tau}. \quad (25) $$
+
+Finally, by taking the Fourier transforms of both sides of the above equation and using the definition in Eq. (1), we obtain the frequency-domain correlation function $W_s(\rho_1, \phi_1; \rho_2, \phi_2; \omega_s)$ for the signal photon:
+
+$$ W_s(\rho_1, \phi_1; \rho_2, \phi_2; \omega) = \sum_{l_s, p_s, p'_s} C_{l_s p_s p'_s}(\omega) [\mathrm{LG}_{p'_s}^{*l'_s}(\rho_1, \phi_1)] \\ \times [\mathrm{LG}_{p'_s}^{l'_s}(\rho_2, \phi_2)]. \quad (26) $$
+
+We see at once that the correlation function for the signal photon has the same functional form as the cross-spectral-density function considered in Eq. (7). Therefore, it follows that, with respect to a detection system that is sensitive only to the azimuthal mode index, the angular coherence function for the signal photon has the same functional form as that of $W(\phi_1, \phi_2)$ in Eq. (16). From now on, we suppress the frequency argument in writing the correlation functions for the signal photon. Starting from Eq. (26) and using the procedure of Sec. II B, one can show that the angular coherence function $W_s(\phi_1, \phi_2)$ corresponding to the signal photon is
+
+$$ W_s(\phi_1, \phi_2) = \sum_{l_s = -\infty}^{\infty} C_{l_s} e^{-il_s \Delta\phi}, \quad (27) $$
+
+where $C_{l_s} = (1/2\pi) \sum_{p_s} C_{l_s p_s p'_s}$. We note that the signal photon field is an incoherent superposition of modes carrying different OAMs. Also, from our discussions in the previous section, we find that if $C_{l_s}$ has a broad distribution in $l_s$, its width can be measured directly by measuring the coherence angle of the signal field. This fact has one very important implication which we discuss in the next section.
+
+B. The angular Schmidt spectrum and the coherence angle
+
+A complete characterization of OAM entanglement of the two-photon state shown in Eq. (21) can be performed through Schmidt decomposition, which yields the Schmidt modes, a natural set of biorthogonal mode pairs that constitute the two-photon state [23]. We note that the two-photon state of Eq. (21) is not in the Schmidt-decomposed form since therein we have summation over three different indices. However, in many quantum-information protocols, such as those based on OAM entanglement, one is concerned with only the OAM-mode index of the photons. In such cases, the detection system is sensitive only to the OAM-mode index and therefore the two-photon state of Eq. (21) can be written in a Schmidt decomposed form that is only one dimensional:
+
+$$ |\psi_2\rangle = \sum_{l=-\infty}^{\infty} \sqrt{C_l} |l\rangle_s |-l\rangle_i. \quad (28) $$
+
+Here $s$ and $i$ stand for signal and idler photons, respectively, and $|l\rangle$ represents an eigenmode of order $l$, corresponding to an azimuthal phase $e^{il\phi}$. $C_l$ is the angular Schmidt coefficient, which is the probability that the signal and idler photons are generated in modes of order $l$ and $-l$, respectively. The distribution of this mode probability is referred to as the angular Schmidt spectrum or the spiral spectrum of the PDC photons [13,24]. For the two-photon state in Eq. (28),

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+the corresponding angular coherence function of the signal
+photon is still given by Eq. (27). The angular Schmidt
+spectrum is directly related to the entanglement of the two-
+photon field through an entanglement measure known as the
+Schmidt number K, which for the normalized angular Schmidt
+spectrum is defined as [23,25,26]:
+
+$$
+K \equiv \frac{1}{\sum_{l=-\infty}^{\infty} C_l^2}. \qquad (29)
+$$
+
+There are two generic ways in which the angular Schmidt
+spectrum of the two-photon field can be measured. First is by
+directly measuring the two-photon intensity in coincidence
+at different values of the OAM-mode index [11,12], and
+the second is by using the Hong-Ou-Mandel interference
+technique [13]. However, comparing Eqs. (28) and (27), we
+find that the OAM-mode spectrum of the signal photon is
+identically equal to the angular Schmidt spectrum of the
+two-photon field. Therefore, it follows that by measuring the
+angular coherence function, as shown in the previous section,
+of the signal field, one can construct the angular Schmidt
+spectrum of the two-photon field. We note that in this method
+one calculates the angular Schmidt (spiral) spectrum without
+doing coincidence measurements, in contrast to the above-
+mentioned methods, which require coincidence detection. In
+situations in which $C_l$ has a broad, Gaussian distribution, it can
+be shown that $K \approx 2\sqrt{\pi}\sigma$, where $\sigma$ is the standard deviation
+of the distribution. This approximate equality becomes an
+exact equality in the limit in which the distribution becomes
+infinitely broad. Now, for the Gaussian distribution, $\sigma$ is
+equal to $1/\phi_{\text{coh}}$, where $\phi_{\text{coh}}$ is the coherence angle of the
+beam. Thus the Schmidt number in this case is inversely
+proportional to the coherence angle of the signal or the idler
+field: $K \approx 2\sqrt{\pi}/\phi_{\text{coh}}$. We thus find that as the entanglement
+of the two-photon field increases the coherence angle of the
+signal and idler fields decreases. We note that the above method
+of estimating the entanglement of the two-photon field by
+performing measurements on the one-photon signal or idler
+field is applicable only when the two-photon field can be
+assumed to be in the Schmidt decomposed form of Eq. (28) and
+fortunately the field produced by the down-converter does have
+this form. In situations in which this assumption is not valid,
+the entanglement has to be estimated by doing measurements
+on the entire two-photon field.
+
+IV. CONCLUSIONS
+
+In conclusion, we have studied partially coherent fields
+that have coherent-mode representations in the OAM-mode
+basis. We have introduced the concepts of the angular
+correlation function and the coherence angle, and by utiliz-
+ing the concept of partial angular coherence we have also
+proposed a method to measure the angular Schmidt spectrum
+of the entangled two-photon field produced by parametric
+down-conversion. This proposed method may have important
+implications as it requires only singles measurement, as
+opposed to the other methods which are based on coincidence
+measurements.
+
+ACKNOWLEDGMENTS
+
+We gratefully acknowledge financial support through a
+MURI grant from the US Army Research Office and by the
+DARPA InPho program through the US Army Research Office
+Award No. W911NF-10-1-0395.
+
+APPENDIX A: ANGULAR COHERENCE OF FIELDS WITH A BROAD DISTRIBUTION FOR C<sub>l</sub>
+
+In this Appendix, we consider partially coherent fields with a broad distribution for $C_l$. For such fields, we can calculate the exact functional form of the angular correlation function $W(\phi_1, \phi_2)$. First, we write the angular correlation function in the following form:
+
+$$
+\begin{align}
+W(\phi_1, \phi_2) &= \frac{1}{2\pi} \sum_{l=-\infty}^{\infty} C_l e^{-il\Delta\phi} \\
+&= \frac{1}{2\pi} \int_{-\infty}^{\infty} C(l) \mathrm{comb}(l) e^{-il\Delta\phi} dl, \quad (\text{A1})
+\end{align}
+$$
+
+where $C(l)$ is a continuous function of $l$. The comb function is defined as $\text{comb}(l) = \sum_{n=-\infty}^{\infty} \delta(l-n)$. In the above equation, the angular correlation function $W(\phi_1, \phi_2)$ is, up to a constant, the Fourier transform of the product of $C(l)$ and $\text{comb}(l)$. We can therefore write it as the convolution of the Fourier transforms of $C(l)$ and $\text{comb}(l)$, that is,
+
+$$
+\begin{align*}
+W(\phi_1, \phi_2) &= \frac{1}{\sqrt{2\pi}} \mathcal{F}[C(l)] \otimes \mathcal{F}[\text{comb}(l)] \\
+&= \frac{1}{\sqrt{2\pi}} C(\Delta\phi) \otimes \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \sum_{n=-\infty}^{\infty} \delta(l-n) e^{-il\Delta\phi} dl \\
+&= \frac{1}{2\pi} C(\Delta\phi) \otimes \sum_{n=-\infty}^{\infty} e^{-i\Delta\phi n}, \tag{A2}
+\end{align*}
+$$
+
+where $\otimes$ represents the convolution and $\mathcal{F}[\cdots]$ the Fourier transformation; $C(\Delta\phi)$ is the Fourier transform of the OAM-mode distribution $C(l)$. Using the formula $\sum_n e^{-i\Delta\phi n} = 2\pi \sum_k \delta(\Delta\phi - 2\pi k)$, we write $W(\phi_1, \phi_2)$ as
+
+$$
+\begin{align}
+W(\phi_1, \phi_2) &= C(\Delta\phi) \otimes \sum_{k=-\infty}^{\infty} \delta(\Delta\phi - 2\pi k) \nonumber \\
+&= \sum_{k=-\infty}^{\infty} C(\Delta\phi - 2\pi k). \tag{A3}
+\end{align}
+$$
+
+Now we restrict the values of Δφ to be between 0 and 2π and assume that the width of C(Δφ) is much smaller than π. This is justified as we have already assumed that C_l has a broad distribution. We thus find that the only significant contribution to W(φ₁, φ₂) comes from the k = 0 term, and thus we obtain W(φ₁, φ₂) = C(Δφ), that is, the angular correlation function is the Fourier transform of the OAM-mode distribution. In the case in which C(l) is Gaussian, that is, C(l) = 1/((√2πσ) exp[-l²/(2σ²)]), where σ is the standard deviation of the distribution, the degree of coherence is given by λ(Δφ) = exp(-σ²Δφ²/2).

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+# APPENDIX B: EVALUATION OF THE SUMMATION IN Eq. (12)
+
+Rearranging the equation
+
+$$ \sum_{p} [\mathrm{LG}_{p}^{l}(\rho)][\mathrm{LG}_{p}^{*l}(\rho')] = \sum_{p} \frac{2p!}{\pi(|l|+p)!} \frac{1}{w_{0}^{2}} \left(\frac{2\rho\rho'}{w_{0}^{2}}\right)^{|l|} \exp\left(-\frac{\rho^{2} + \rho'^{2}}{w_{0}^{2}}\right) L_{p}^{|l|}\left(\frac{2\rho^{2}}{w_{0}^{2}}\right) L_{p}^{|l|}\left(\frac{2\rho'^{2}}{w_{0}^{2}}\right), \quad (B1) $$
+
+we can write the above equation as
+
+$$ \sum_p [\mathrm{LG}_p^l(\rho)][\mathrm{LG}_p^{*l}(\rho')] = \frac{2}{\pi w_0^2} \left(\frac{2\rho\rho'}{w_0^2}\right)^{|l|} \exp\left(-\frac{\rho^2 + \rho'^2}{w_o^2}\right) \sum_p \frac{\Gamma(p+1)}{\Gamma(|l|+p+1)} L_p^{|l|}\left(\frac{2\rho^2}{w_0^2}\right) L_p^{|l|}\left(\frac{2\rho'^2}{w_0^2}\right). \quad (B2) $$
+
+The summation on the right-hand side is a standard result for Laguerre polynomials, using which we get
+
+$$ \sum_{p} [\mathrm{LG}_{p}^{l}(\rho)][\mathrm{LG}_{p}^{*l}(\rho')] = \frac{2}{\pi w_{0}^{2}} \left(\frac{2\rho\rho'}{w_{0}^{2}}\right)^{|l|} \exp\left(-\frac{\rho^{2}+\rho'^{2}}{w_{o}^{2}}\right) \left(\frac{2\rho^{2} 2\rho'^{2}}{w_{0}^{2} w_{0}^{2}}\right)^{-|l|/2} \exp\left(\frac{\rho^{2}+\rho'^{2}}{w_{0}^{2}}\right) \delta\left(\frac{2\rho^{2}}{w_{0}^{2}} - \frac{2\rho'^{2}}{w_{0}^{2}}\right). \quad (B3) $$
+
+Finally after rearranging, we get the desired result
+
+$$ \sum_p [\mathrm{LG}_p^l(\rho)][\mathrm{LG}_p^{*l}(\rho')] = \frac{1}{\pi} \delta(\rho^2 - \rho'^2). \qquad (B4) $$
+
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+# Error Analysis of the Square Root Operation for the Purpose of Precision Tuning: a Case Study on K-means
+
+Oumaima Matoussi, Yves Durand, Olivier Sentieys, Anca Molnos
+
+► To cite this version:
+
+Oumaima Matoussi, Yves Durand, Olivier Sentieys, Anca Molnos. Error Analysis of the Square Root Operation for the Purpose of Precision Tuning: a Case Study on K-means. ASAP 2019 - 30th IEEE International Conference on Application-specific Systems, Architectures and Processors, Jul 2019, New York, United States. pp.1-8. [hal-02183945](https://hal.inria.fr/hal-02183945)
+
+HAL Id: hal-02183945
+
+https://hal.inria.fr/hal-02183945
+
+Submitted on 15 Jul 2019
+
+**HAL** is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
+
+L'archive ouverte pluridisciplinaire **HAL**, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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+# Error Analysis of the Square Root Operation for the Purpose of Precision Tuning: a Case Study on K-means
+
+Oumaima Matoussi, Yves Durand
+
+CEA, LETI, Univ. Grenoble Alpes, France
+name.sumame@cea.fr
+
+Olivier Sentieys
+
+Inria, Univ. Rennes, France
+olivier.sentieys@inria.fr
+
+Anca Molnos
+
+CEA, LETI, Univ. Grenoble Alpes, France
+anca.molnos@cea.fr
+
+**Abstract**—In this paper, we propose an analytical approach to study the impact of floating point (FLP) precision variation on the square root operation, in terms of computational accuracy and performance gain. We estimate the round-off error resulting from reduced precision. We also inspect the Newton Raphson algorithm used to approximate the square root in order to bound the error caused by algorithmic deviation. Consequently, the implementation of the square root can be optimized by fittingly adjusting its number of iterations with respect to any given FLP precision specification, without the need for long simulation times. We evaluate our error analysis of the square root operation as part of approximating a classic data clustering algorithm known as K-means, for the purpose of reducing its energy footprint. We compare the resulting inexact K-means to its exact counterpart, in the context of color quantization, in terms of energy gain and quality of the output. The experimental results show that energy savings could be achieved without penalizing the quality of the output (e.g., up to 41.87% of energy gain for an output quality, measured using structural similarity, within a range of [0.95,1]).
+
+**Index Terms**—approximate computing, error analysis, round-off error, algorithmic deviation, square root, Newton Raphson method, precision tuning, k-means, clustering, floating point
+
+## I. INTRODUCTION
+
+The ever increasing volume, diversity and high dimensional- ity of data goes hand in hand with the rapid growth of energy consumption of computer systems. Thus, improving the energy efficiency of computer systems that try to keep pace with the constant growth of information is a critical concern.
+
+In many application domains that deal with huge amounts of data such as multimedia processing (images, audio, video, etc.), data mining and machine learning, computations can be tolerant to some degree of error without critical degradation in the quality of the output. For example, a small image quality loss due to some modification of the color of a group of pixels can be hardly noticed by the user due to the limited capabilities of human perception. In such classes of applications, output accuracy could be traded for energy reduction.
+
+Performance gains can be achieved at the application level using software techniques, thanks to a relatively new comput- ing paradigm called *approximate computing* [1]. Introducing inexactness in computations may lead to energy scaling with little to no loss in accuracy. A way to introduce inexactness in an application is by precision reduction of FLP variables
+
+and computations [2], [3]. Finding the sweet spot between the numerical accuracy and the performance of the application (e.g. energy consumption) by refining the bit-width of the FLP variables is known as *precision tuning*.
+
+A loss in computational accuracy is inevitable due to reduced precision. However, to be able to determine the optimal precision (i.e. bit-width) of FLP variables that minimizes energy cost with the least impact on computational accuracy, two approaches can be considered: FLP simulation and analytical techniques to track round-off error (i.e. error introduced due to limited precision). FLP simulation consists in using libraries (e.g. MPFR) that allow the definition of FLP variables with adjustable bit-widths and provide the appropriate arithmetic operations. Nonetheless, the search space of the optimal bit- width can be very large, in that every change in the format of a FLP variable requires a new simulation, which can be very time consuming.
+
+Analytical approaches, on the other hand, try to determine a mathematical formula that models the impact of quantization error on the accuracy of the output [4], [5]. Once this formula is established, it can be applied to any application with different FLP formats. Unfortunately, this only works with smooth operations (arithmetic operations like addition, subtraction, multiplication, division, etc.). So, analytical approaches help save the time spent on tuning FLP precision but fail to estimate the computational error in the presence of non-smooth operations. In this case, simulation becomes inescapable. Thus, we advocate through our work the combination of simulation and analytical techniques.
+
+A plethora of work using simulation-based approaches for the purpose of FLP precision tuning was propounded [3], [6], [7], etc. As for analytical approaches, efforts were made in the context of fixed-point word-length optimization [4], [8] and mainly focused on common arithmetic operations like addition and multiplication. Square root operation is significantly more costly, in terms of energy consumption, than addition or multiplication and requires several iterative cycles to complete. Moreover, unlike simple arithmetic operations, the square root is usually implemented using the *Newton Raphson* method, which is an approximation of the operation itself, and adds another type of error referred to as *algorithmic deviation* besides the round-off error. Consequently, error analysis of a square

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+root operation is not a straightforward task and entails a two-fold method.
+
+The main contribution of this paper consists in conducting static error analysis of the square root operation and estimating a bound on the errors caused by limited precision and algorithmic deviation. Based on the formally derived error bound, the number of iterations of the square root operation is automatically adjusted for each precision. Therefore, the implementation of the square root (in terms of number of iterations) can be optimized with respect to any given FLP precision specification without the need for long simulation times.
+
+In addition to the stand-alone analysis of the square root operation and for a more complete evaluation of our proposition, the second contribution of this paper consists in evaluating our analytical approach by studying the square root operation as part of a whole application. Our choice fell on a data clustering algorithm called K-means a.k.a. Lloyd's algorithm [9]. It is a type of unsupervised learning algorithm used to cluster a set of unlabeled data into $k$ clusters based on data feature similarity [10]. The similarity is typically determined using the Euclidean distance measure, which involves square root operations. The distance function is the mainstay of the majority of clustering and classification algorithms, which will help us highlight the impact of square root optimization on the performance of the application as a whole.
+
+It should be noted that the proposed analytical approach for the square root operation is independent of the application itself. Any clustering algorithm or any other application (e.g. digital signal and image processing, 3D graphics, spectrum analysis, wireless communications, etc.), for that matter, where square root computations can be found, would also benefit from our error analysis method. The importance of our contribution lies within the integration of our analytical approach, which is aimed for a smooth operation, namely the square root, in an application that also contains non-smooth operations. Hence, the combined effort of simulation and analytical results is needed for precision tuning. We quantify the efficiency of the proposed approach by measuring the quality degradation of K-means' output as well as its energy gain.
+
+The rest of this paper is organized as follows. After overviewing contributions dealing with approximate computing in Section II, we detail our error analysis approach of the square root operation in Section III. We explain the process of sensitivity analysis of K-means, based on our square root analysis results and variable-precision FLP simulation in Section IV. We validate the efficiency of approximate K-means by measuring both SSIM and energy gain and we also discuss the experimental results in Section V, before concluding the paper in Section VI.
+
+## II. RELATED WORK
+
+Precision tuning is a research direction that has received significant attention and is motivated by the fact that FLP operations contribute to the energy footprint of an application.
+
+In [6], a framework called ASAC using statistical methods was proposed. This framework helps discover approximable and non-approximable program parts through sensitivity analysis. The main idea is to perturb program variables and check the resultant output against a correct output, i.e. one fulfilling an acceptable QoS threshold.
+
+Precimonious [7], a dynamic program analysis tool, was proposed to assist developers in choosing the lowest precision that satisfies accuracy and performance constraints. Given a set of FLP variables and their possible types, a search based on the delta-debugging algorithm is performed. The search outputs a type configuration that maps each variable to a type. However, Precimonious requires a representative set of program inputs provided by the user. If the same type configuration is applied on a much worse conditioned input then no guarantees can be made.
+
+The method proposed in [2] is not limited to determining the best mix of single or double precision in a program as in [7], but it computes the precision of FLP variables down to the bit level of the mantissa. A heuristic precision tuning algorithm based on a binary search is performed in order to find the smallest precision possible for each variable while keeping the output error within some user-given bound. The results show that the speedup gained by tuning some programs may be diminished or even eliminated because of the overhead due to data type conversions performed by the compiler.
+
+In [3], hardware implementations of fixed-point and FLP arithmetic operators are compared in terms of area, delay and energy. A custom FLP library called `ct_float` was devised to vary the bit width of the variables. The authors concluded that FLP operators provide a better energy/accuracy tradeoff for small bitwidths but important area, delay and energy overhead for larger bitwidths, compared to fixed-point operators.
+
+The common denominator in the majority of FLP precision tuning approaches is that they rely on varying the precision of the different FLP variables in an application (usually with the help of a multiple precision FLP library) and testing different combinations before deciding on the optimal precision. This process is prohibitively expensive, time wise. Furthermore, when big applications with a huge set of input is under optimization, the search space for optimal precision can be enormous and it might be impossible to cover fully.
+
+Analytical approaches are faster than simulation because once a mathematical expression that models (or bounds) the round-off error is determined, it can be applied to different precisions. So, it is a one-time effort whose results can be applied to multiple FLP formats. Efforts have been made to find analytical models in the context of accuracy evaluation of fixed-point systems. A series of propositions of analytical methods to measure the output noise of signal processing systems were presented in [4], which dealt with smooth operations, then in [5], which catered for decision operators and recently in [8], where a hybrid approach based on both simulation and analytical results was highlighted.
+
+A parametric error analysis of a version of Goldschmidt's square root algorithm that uses directed rounding implemen-

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+tations, was proposed in [11]. The error analysis is based on
+relative errors of intermediate computations. The proposed
+error formulae were intended to help determine the optimal
+hardware implementation (i.e. multiplier dimensions). However,
+their analysis was not demonstrated on a real architecture.
+
+In our work, we study analytically the square root operation,
+which is the backbone of many algorithms such as clustering,
+data mining and signal processing algorithms, with respect to
+FLP precision tuning. Moreover, we quantify the efficiency
+of the proposed error bound in the context of K-means by
+comparing the QoS of the inexact version of K-means to its
+exact counterpart. We also measure relative energy gains using
+energy costs obtained from [3], [12] and profiling information.
+
+### III. ANALYSIS OF THE SQUARE ROOT OPERATION
+
+First, to eliminate any confusion, it should be noted that the
+terms accuracy and precision are not used interchangeably in
+this paper. Nearly all processors and programming languages
+support FLP numbers, which are defined in the IEEE-754
+normalization. A FLP number *f* is represented by an exponent
+*e*, a mantissa *m* and a sign bit *s*: $f = (-1)^s \times m \times 2^e$. We
+designate by precision the number of bits used to represent the
+mantissa of a FLP variable, whereas we refer by accuracy to
+the quality of the result, i.e., the degree of error in the output,
+compared to a reference or a *golden* output.
+
+The square root operation is usually approximated with the *Newton Raphson* method. To compute $y = \sqrt{a}$, $a > 0$, the Newton Raphson method starts with an initial guess (i.e. initial seed value) $y_0 > 0$. The initial guess is then refined by iterating over:
+
+$$y_{n+1} = \frac{1}{2} \left( y_n + \frac{a}{y_n} \right)$$
+
+However, this conventional iteration for the square root computation is not frequently used because it entails a division at each step. Since division is generally much slower than multiplication, the square root reciprocal is usually advocated. The square root reciprocal converges to $\frac{1}{\sqrt{a}}$ and iterates over:
+
+$$x_{n+1} = \frac{x_n}{2} \times (3 - a \times x_n^2).$$
+
+In order to get $\sqrt{a}$, the result is multiplied by *a*.
+
+To optimize the implementation of the Newton Raphson method, we aim at adjusting the number of its iterations according to the selected precision, without jeopardizing the accuracy of the result. To do so, we perform error analysis to statically determine at which iteration it is preferable to stop the computations, for a specific precision *p*, without causing further error. The two main causes of error that we investigated are:
+
+• the round-off error caused by FLP representation and FLP operations,
+
+• and the systematic error (a.k.a. unavoidable error or algorithmic deviation), which results from the Newton Raphson approximation itself.
+
+#### A. Bounding the Round-off Error
+
+Let $fl(x+y) = (x+y)(1+\epsilon_{add})$ and $fl(x\times y) = (x\times y)(1+\epsilon_{mul})$, be approximations of the exact mathematical operations
+
++ and × respectively, $\forall x, y \in \mathbb{R}$, such that $|\epsilon_{add,mul}| \le \epsilon_m$.
+$\epsilon_m$, called machine epsilon or the unit roundoff, is defined as
+the smallest number such that $1 + \epsilon_m > 1$ [13]. $\epsilon_m = \beta^{-p}$,
+where $\beta$ is the base and $p$ is the number of bits used for the
+magnitude of the mantissa of a FLP number represented as
+$r_0.r_1r_2...r_{p-1} \times \beta^e$. We assume that $\epsilon_{add} = \epsilon_{mul} = \epsilon$ and that
+multiplication by $\frac{1}{2}$ does not incur round-off error.
+$\forall x_i \in \mathbb{R}$, let $\hat{x}_i = fl(x_i)$ be the FLP representation of $x_i$;
+$i = 1..n+1$. Machine epsilon is an upper bound on the relative
+error in representing a FLP number: $\frac{|\hat{x}_i-x_i|}{|x_i|} \le \epsilon_m$.
+We would like to bound the round-off error of the Newton
+Raphson reciprocal. To do so, we aim at expressing $\hat{x}_{n+1}$ as:
+$\hat{x}_{n+1} \le x_{n+1} \times (1 + \delta)$, where $\delta$ is the round-off error.
+
+$$ 
+\begin{gather*}
+\hat{x}_{n+1} = fl(x_{n+1}) = fl\left(\frac{\hat{x}_n}{2} \times fl\left(3 - fl(a \times fl(\hat{x}_n^2))\right)\right) \\
+\hat{x}_{n+1} = \frac{\hat{x}_n}{2} \times \left(3 - a \times \hat{x}_n^2(1+\epsilon)^2\right)(1+\epsilon)^2 \quad (1)
+\end{gather*}
+$$
+
+Let $e(\hat{x}_n) = 3 - a \times \hat{x}_n^2 (1 + \epsilon)^2$.
+
+$$|e(\hat{x}_n)| \leq |3 - a \times \hat{x}_n^2| + |2a \times \hat{x}_n^2 \times \epsilon| \quad (2)$$
+
+We have:
+
+$$\sqrt{a} \times x_n < 1 \quad (3)$$
+
+*Proof.* $x_{n+1}\sqrt{a}-1 = -(1+x_n\frac{1}{2}\sqrt{a})(x_n\sqrt{a}-1)^2 < 0$ ☐
+
+Based on Equation 3 we can conclude that
+
+$$\epsilon < \frac{(3 - a \times x_n^2) \times \epsilon}{2}. \quad (4)$$
+
+Then, combining Equations 2 and 4 gives:
+
+$$|e(x_n)| < (3 - a \times x_n^2) \times (1 + \epsilon), \quad (5)$$
+
+and combining Equations 1 and 5 gives
+
+$$\hat{x}_{n+1} < \frac{\hat{x}_n}{2} \times (3 - a \times \hat{x}_n^2) \times (1 + \epsilon)^3.$$
+
+Disregarding $O(\epsilon^n)$ terms, $n > 1$, in the previous inequal-
+ity [13], we obtain
+
+$$\hat{x}_{n+1} < \frac{\hat{x}_n}{2} \times (3 - a \times \hat{x}_n^2) \times (1 + 3\epsilon), \quad (6)$$
+
+from which we can conclude that the round-off error is $3\epsilon$ per
+iteration. Note that $\hat{x}_{n+1}$ converges towards $\frac{1}{\sqrt{a}}$ and that we
+want to bound the error of $\hat{y}_{n+1}$, which converges towards $\sqrt{a}$.
+So, only the last iteration should be multiplied by $a$:
+
+$$ 
+\begin{aligned}
+x_{n+1} \times a &= y_{n+1} &\implies fl(x_{n+1} \times a) &= fl(y_{n+1}) \\
+& & &\implies \hat{x}_{n+1} \times a \times (1 + \epsilon) &= \hat{y}_{n+1}.
+\end{aligned}
+$$
+
+Consequently, the overall round-off error is $\delta \le n \times 3\epsilon + \epsilon$.
+
+#### B. Bounding the Systematic Error
+
+The absolute systematic error in computing $\frac{1}{\sqrt{a}}$ is
+
+$$|x_i - \frac{1}{\sqrt{a}}|.$$

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+Fig. 1: Systematic and round-off errors for different precision values and iteration values of Newton Raphson
+
+The relative systematic error in computing $\frac{1}{\sqrt{a}}$ is
+
+$$ \frac{|x_i - \frac{1}{\sqrt{a}}|}{\frac{1}{\sqrt{a}}} = |\sqrt{a}x_i - 1|. $$
+
+To find an upper bound for the systematic error, we start with a reasonable initial value defined as
+
+$$ \frac{1}{2} \times \frac{1}{\sqrt{a}} \le x_0 \le \frac{3}{2} \times \frac{1}{\sqrt{a}} \Rightarrow 1 + 0.5 \times \sqrt{a} \times x_0 \le \frac{7}{4} \text{ and } \sqrt{a} \times x_0 - 1 \le \frac{1}{2}. $$
+
+So, according to the proof of Equation 3:
+
+$$ |\sqrt{a} \times x_1 - 1| \le \frac{4}{7} \times \left| \frac{7}{4} \times (\sqrt{a} \times x_0 - 1) \right|^2. $$
+
+Since $\frac{7}{4} \times (\sqrt{a} \times x_0 - 1) \le \frac{7}{8}$ then, $|\sqrt{a} \times x_1 - 1| \le \frac{1}{2} \times \frac{7}{8}$.
+
+It is safe to conclude that the systematic error in computing $\frac{1}{\sqrt{a}}$ at iteration $i=0..n$ is
+
+$$ |\sqrt{a} \times x_i - 1| \le \frac{1}{2} \times (\frac{7}{8})^{2^i-1}. \quad (7) $$
+
+The relative systematic error for $\frac{1}{\sqrt{a}}$ is the same as the one for $\sqrt{a}:$
+
+$$ \frac{|y_i - \sqrt{a}|}{\sqrt{a}} = \frac{|ax_i - \sqrt{a}|}{\sqrt{a}} = |\sqrt{a}x_i - 1|. $$
+
+Both the systematic and round-off errors are plotted for different precision values ($p$) in Fig. 1. For space reason, only two of the resulting graphs are illustrated. The straight line corresponds to the round-off error, which increases proportionally to the number of iterations of the *Newton Raphson* method, whereas the curve designates the systematic error, which declines as the number of iterations grows.
+
+The rise of the round-off error is remarkably rapid at lower
+
+precision values (e.g. Fig. 1(a)) but it slows down at higher precision (e.g. Fig. 1(b)).
+
+The systematic error, on the other hand, is independent of the precision. It stabilizes at zero around the sixth iteration. This means that, disregarding the round-off error (round-off error equal to 0), six *Newton Raphson* iterations are sufficient in producing accurate (i.e. error=0) square root approximation. However, when the round-off error is in the picture, which is usually the case, the higher the number of iterations, the higher the round-off error is. Although at six sqrt iterations the systematic error is 0, the round-off error is 0.6 for $p = 4$ and 0.05 for $p = 9$.
+
+The round-off error accumulates as the number of iterations rises, which counteracts the decline of the unavoidable error. That is why, the round-off error should not exceed the systematic error. Thus, the intersection between the two lines is indicative of the *optimal* number of iterations at which we have a *good enough* result.
+
+## IV. APPLICATION TO K-MEANS
+
+So far, we studied the square root operation individually, in the context of error analysis for the purpose of precision tuning. It would be interesting to apply our study of the square root in a fullblown application that encompasses both smooth and non-smooth operations. We chose a clustering algorithm called K-means that uses the square root operation in the computation of the Euclidean distance function.
+
+In this section, we start by describing K-means and its use in color quantization. Then, we present precision tuning of K-means that involves both FLP simulation and our analytical results.
+
+### A. Color Quantization Using K-Means
+
+The basic principle of the clustering problem is as follows. A data set $X = (x_1, x_2, ..., x_n)$ is composed of data elements, where each data element $x_i$ is a $d$-dimensional vector. This data set is partitioned into $k$ clusters of similar data points [14].
+
+In the particular case of color quantization, one of K-means' applications, an image represented as an array of size $N \times d$ where $N$ is the number of pixels in the image and $d$ is the color space (referred to as data dimension or data features), is represented with a smaller number of colors. Most commonly, the space is 3-dimensional ($d=3$) and the coordinates encode the color. For example, RGB is a color space usually encoded as a 3-tuple of 8 bits each. The value of each dimension is within the range of [0, 255].
+
+The flowchart in Fig. 2 gives an overview of the different steps of the clustering process in the context of color quantization with K-means.
+
+The goal of the K-means clustering algorithm is to generate a compressed image out of the original image. To this end, a *palette*, i.e. a set of centroids $C = (c_1, c_2, ..., c_k)$, is firstly chosen by selecting data elements (i.e. colors or pixels) that best represent the original image, for each cluster. In other words, the number of centroids is the number of colors that the palette is made of. Then, each data element is mapped to the *closest*

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+Fig. 2: Flowchart of the K-means algorithm
+
+color in the palette based on a distance measure $||x||_m = (\sum_{i=1}^n |x_i|^m)^{1/m}$ (e.g. $m=1$ for the Manhattan distance, $m=2$ for the Euclidean distance) [15]. This results in a preliminary classification. Following this initial classification, $k$ new centroids are re-computed as barycenters of the previously generated clusters and data elements are re-assigned to the new centroids based on the chosen distance measure. The algorithm is based on an iterative refinement technique and it iterates until centroids do not move anymore (i.e., no data elements change clusters), a maximum number of iterations is reached or the distance measure is minimized, as illustrated in Fig. 2. To quantify the difference between two pixels, a simpler distance function known as the Manhattan distance (sum of absolute differences) is usually used. However the optimal distance, in terms of minimizing within-class variance and producing higher-quality clusters, and more complicated, in terms of computational complexity, is the Euclidean distance (sum of the squared distances) [16]. Unlike other studies on approximate K-means, we use the optimal distance metric:
+
+$$ dist(p1, p2) = \sqrt{(R1 - R2)^2 + (G1 - G2)^2 + (B1 - B2)^2}. $$
+
+## B. Precision Tuning Using a Multiple-Precision Floating Point Library
+
+We focus our precision analysis on the Euclidean distance function, which is the kernel of the K-means algorithm as it is in charge of computing the distance between a given data point and a cluster centroid. These distance values are pivotal in the correct assignment of pixels to their closest cluster and thus in providing a correct classification. We are going to include approximation in the FLP computations of the Euclidean distance function and leverage the results from our analysis in Section III to account for the impact of FLP precision variation on the square root operation.
+
+Introducing inexactness by reducing the number of bits of the mantissa can sometimes provide results of the same accuracy but with better energy efficiency than the exact version of the program. To this aim, we studied the sensitivity of K-means by arbitrarily varying the number of bits of the mantissa (2-23 bits) and observing the repercussions on the output.
+
+As was established in our analytical study of the square root operation (Section III), the number of bits of the mantissa has an impact on the number of square root iterations. So, to study the sensitivity of K-means, we re-wrote K-means' Euclidean distance function using the MPFR library, which is a smooth extension of the IEEE-754 standard where any FLP number can have its own precision [17], and we assigned to each precision its corresponding number of square root iterations according to the analysis in Section III. Our analytical results come in handy in the precision tuning of K-means in that, we do not have to vary the number of square root iterations for each precision simulation in order to determine the optimal pair $(p, sqrt)$. The optimal number of square root iterations is statically determined for each precision (Section III). For example, for a precision $p=4$, the FLP variables in K-means' Euclidean distance function are transformed into MPFR variables ($mpfr_t var$), their precision is set to $4$ ($mpfr_init2(var, 4)$) and the number of iterations in the Newton Raphson method implementation is set directly to $sqrt = 2$.
+
+In addition to the FLP variables, we tracked the operations that use these variables and changed them into MPFR operations. The Euclidean distance function encompasses multiplication, addition, subtraction and square root operations.
+
+### Listing 1: MPFR operations
+
+```c
+mpfr_sub(r, r, r1, MPFR_RNDN); /* r=r-r1; */
+mpfrMul(r, r, r, MPFR_RNDN);   /* r=r*r; */
+mpfr_add(r, r, r2, MPFR_RNDN);  /* r=r+r2; */
+mpfr_Sqrt(r_tmp, r, MPFR_RNDN); /* r_tmp=sqrt(r) */
+```
+
+Listing 1 showcases the transformation of different operations into MPFR operations. We implemented two inexact versions of the Euclidean distance function. We called these versions fused and unfused. In the first, the *sub*, *mul* and *add* operations are computed with full precision and the result is rounded once to *N* significant bits and then passed to the reduced-precision square root function. In the latter, all the operations are performed with reduced-precision, which means that the value is rounded four times before yielding the final result. We examine both fused and unfused versions in Section V to determine which version is more beneficial to implement.
+
+The quality of the result as well as the energy consumption depend not only on the precision $p$ of the FLP variables but also on other parameters such as the number of clusters $k$ and the number of iterations $n$ needed for the clustering process to converge. More clusters means that the compressed image will contain more colors, which is in favor of a better quality. With more iterations, there is a better chance for K-means to converge, which also contributes to the quality of the output. So, we investigated precision variation and its impact on the

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+square root iterations over a range of different values of (k, n) pairs. Thus, we also varied the (k, n) parameters along with the precision and quantified the impact on the output in order to discern the best configuration that yields the desired energy-QoS tradeoff.
+
+V. EXPERIMENTAL RESULTS
+
+To determine whether a configuration (k, n) for a given
+precision p is favorable, we transform the original source code
+of K-means [18] using the configuration in question, compile it
+to an ARM binary, execute it while using energy and instruction
+counters, and check for two criteria: energy consumption and
+QoS.
+
+A. Experimental Setup
+
+In order to quantify the impact of approximation on energy
+consumption, we make use of an energy model based on energy
+measurements of ARM Cortex-A7 instructions at a frequency
+of 500 MHz and without dependencies between instructions
+[12]. We also make use of profile information, i.e., instruction
+types (e.g., add, mul, div) and the number of execution of
+each type. ARM's Cortex-A7 core is a dual issue in-order core
+that consists of one load/store, one multiply, one FLP and two
+integer units. A thorough characterization of the ARM Cortex-
+A7 instruction set with energy metrics for every instruction
+type could be found in [12].
+
+The energy values of integer and FLP instructions are
+normalized with reference to integer multiplication, i.e., integer
+multiplication is used as a unit of measure throughout the
+experiments. Normalized energy values of 32-bit integer and
+FLP instructions are listed in Table I. These normalized energy
+values are used to compute the energy consumption of exact
+K-means, which is considered as a touchstone. As for the
+energy values of variable precision (2-16 bits) operations,
+namely addition, multiplication and division, they are computed
+according to [3] and shown in Fig. 3. It is noticeable from
+Fig. 3 that energy consumption is influenced by the number
+of bits of the mantissa; the higher the precision the higher the
+energy cost.
+
+The overall energy consumption is computed by combining
+the profile information and the energy values
+
+$$E_{total} = \sum_{i=1}^{\#types} op_i \times e_i,$$
+
+where *op*<sub>*i*</sub> is the number of operations of type *i* and *e*<sub>*i*</sub> is the energy consumed per operation of type *i* (normalized to *multi*). We estimate the energy savings by comparing the energy consumption obtained by executing the exact version of K-means (i.e. with the highest precision *p* = 23) to the approximated version.
+
+The quality of the result is measured using the SSIM index.
+SSIM is a perception-based metric that compares two images by
+incorporating a number of terms including contrast, luminance
+and structural information. The degradation of the image quality,
+due to compression with inexact K-means (reduced precision
+
+TABLE I: Energy values (normalized to *muli*) of a sample of integer and 32-bit FLP instructions
+
+<table>
+   <thead>
+    <tr>
+     <td>
+      Instr
+     </td>
+     <td>
+      addi
+     </td>
+     <td>
+      muli
+     </td>
+     <td>
+      si
+     </td>
+     <td>
+      li
+     </td>
+     <td>
+      addf
+     </td>
+     <td>
+      mulf
+     </td>
+     <td>
+      sf
+     </td>
+     <td>
+      lf
+     </td>
+     <td>
+      divf
+     </td>
+    </tr>
+   </thead>
+   <tbody>
+    <tr>
+     <td>
+      Energy
+     </td>
+     <td>
+      1.02
+     </td>
+     <td>
+      1
+     </td>
+     <td>
+      2.41
+     </td>
+     <td>
+      1.79
+     </td>
+     <td>
+      1.16
+     </td>
+     <td>
+      1.16
+     </td>
+     <td>
+      2.34
+     </td>
+     <td>
+      1.92
+     </td>
+     <td>
+      7.80
+     </td>
+    </tr>
+   </tbody>
+  </table>
+
+instr:i:integer instruction, instr:f:FLP instruction
+
+Fig. 3: Normalized energy values for different precisions
+
+and/or parameter variation), is determined with respect to the
+compressed image with the highest precision.
+
+B. Results
+
+In this section, we study the impact of the precision and
+square root iterations (*p*, sqrt) on the quality of the output
+for various *n* (number of iterations needed for K-means to
+converge) and *k* (number of clusters) configurations. To do so,
+we start by jointly varying the precision bits *p* and the number
+of K-means iterations *n*, while fixing the number of clusters
+to *k* = 50, for both unfused and fused operations.
+
+Then, we vary the number of clusters *k* and the precision *p* while fixing the number of K-means iterations to *n* = 10. The SSIM values are averaged over 10 different RGB images chosen from [19]. An SSIM value is within the interval [0, 1], 1 designating the best image quality.
+
+The first set of experiments that we conducted, i.e., SSIM(*p*, *n*) shows that the number of iterations *n* has little to no impact on SSIM for different precisions and for both the fused and unfused cases. This is true for other *k* values as well (10, 100, 500, 1000). Accordingly, the number of iterations *n* is fixed to 10 for the rest of the experiments.
+
+However, by varying the precision with the number of clusters while fixing *n* to 10, a noticeable change in SSIM is observed and the results are reported in Fig. 4. For instance, in case of unfused operations (Fig. 4(a)), SSIM increased from 0.791 with *k* = 10 to 0.969 with *k* = 1000, for *p* = 10. Regarding the fused operations’ case (Fig. 4(b)), the results are almost identical to the unfused case except for *k* = 10, where the unfused version yielded better SSIM values for the different tested precisions, and for *p* = 2 and *k* > 10, where the fused version yielded better SSIM values.
+
+Based on our experiments (SSIM(*p*, *k*) in Fig. 4 and SSIM(*p*, *n*)), fused and unfused operations generate similar

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+Fig. 4: Impact of precision variation and number of clusters on SSIM: for (a) unfused operations and (b) fused operations
+
+SSIM values for the majority of the configurations. Taking into consideration the additional requirements that come with fused operations (i.e., dedicated hardware), which is not always supported by all architectures, as well as a compiler that allows the program to make use of the fused operations, we deem it too big an effort for an insignificant quality enhancement. Consequently, we advocate the use of unfused operations in the context of reduced-precision K-means.
+
+It is also worth mentioning that we conducted the experiments from $p = 2$ to $p = 23$, but no noticeable change was detected starting from $p = 16$, as can be observed in Fig. 4. Thus, the rest of the graphs presented in this paper stop at $p = 16$.
+
+To conclude whether approximating K-means is worthwhile, we also measured the energy gains. Fig. 5 displays the number of executed instructions of the Euclidean distance function per instruction type: integer arithmetic, FLP arithmetic, load/store (both FLP and integer) and other instructions (e.g. branch,
+
+Fig. 5: Executed instructions breakdown for exact K-means ($p = 23$) with different $(k, sqrt)$ configurations
+
+Fig. 6: Energy consumption breakdown for exact K-means ($p = 23$) with different $(k, sqrt)$ configurations
+
+comparison, conversion). The values are generated by applying exact K-means ($p = 23$) while varying the number of clusters and the square root iterations. For the different configurations of $(k, sqrt)$, the FLP arithmetic operations contribute with around 19% of the total number of executed instructions compared to integer arithmetic operations, which represent only 2% when $sqrt = 6$ and almost 3% when $sqrt = 3$.
+
+Fig. 6 shows the energy consumption breakdown for exact K-means with different $(k, sqrt)$ configurations. It is clear that the number of clusters $k$ as well as the number of square root iterations $sqrt$ have a significant impact on the energy footprint of the program. For instance, for the same number of clusters $k = 50$, energy consumption increases by approximately $6 \times 10^9$ from $sqrt = 3$ to $sqrt = 6$.
+
+The percentages of energy savings are plotted against $p$ for different $(k, sqrt)$ configurations in Fig. 7. Each precision is attributed its optimal number of square root iterations according to our analytical study (Section III). For example, for $p = 4$, $sqrt = 2$. The graph indicates that there are energy gains for the different $(p, k, sqrt)$ configurations but with different magnitudes. The energy gain of each configuration $(p, k, sqrt)$ is computed with respect to the reference configuration ($p = 23, k = 100, sqrt = 6$). The smallest gain is observed at $p = 16$ and ($k = 100, sqrt = 6$) with a value of 14.09%. The biggest gain is at $p = 2$ and ($k = 10, sqrt = 1$) with a value of 96.57%. To be able to determine the best energy-QoS tradeoff

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+Fig. 7: Percentage of energy gain of different precisions *p* and with different (*k*, sqrt) configurations
+
+Fig. 8: SSIM of different precisions *p* and with different (*k*, sqrt) configurations
+
+for each precision, SSIM is plotted against *p* in Fig. 8 with the same (*k*, sqrt) configurations as in Fig. 7. The same reference (*p* = 23, *k* = 100, sqrt = 6) also serves as a baseline. The best quality is reached at *p* = 16 and (*k* = 100, sqrt = 6) with an SSIM value of 1. The worst quality is noticed at *p* = 2 and (*k* = 10, sqrt = 1) with an SSIM value of 0.67.
+
+Assuming that an output image with an SSIM value lying within the range of [0.95, 1] is considered a good quality image, then configuring K-means with a 6-bit mantissa, 100 clusters and 4 square root iterations is sufficient in producing such an image while saving 41.87% of energy.
+
+## VI. CONCLUSION
+
+This paper focused on optimizing the implementation of the square root operation. An analytical examination of the Newton Raphson approximation of the square root showed that the number of square root iterations is influenced by the precision bits. Consequently, we associated to each precision its optimal
+
+number of Newton Raphson iterations. For a well-rounded evaluation of our proposition, we aimed at finding opportunities to reduce energy consumption of a classic clustering algorithm called K-means by varying the precision of its FLP variables and adjusting the number of square root iterations of its distance function.
+
+The various approximated versions of K-means were compared to the exact version in terms of QoS, measured with SSIM, and relative energy gain. The obtained results can serve as a guideline in choosing the best configuration of precision bits, number of clusters and square root iterations (*p*, *k*, sqrt), i.e., one that yields the most energy gains, for a desired QoS (e.g., for an SSIM within [0.95, 1], an energy gain of 41.87% is achieved with a (6, 100, 4) configuration).
+
+## REFERENCES
+
+[1] Q. Xu, T. Mytkowicz, and N. S. Kim, "Approximate computing: A survey," *IEEE Design Test*, Feb 2016.
+
+[2] N. M. Ho, E. Manogaran, W. F. Wong, and A. Anoosheh, "Efficient floating point precision tuning for approximate computing," in *ASP-DAC*, Jan 2017.
+
+[3] B. Barrois and O. Sentieys, "Customizing Fixed-Point and Floating-Point Arithmetic - A Case Study in K-Means Clustering," in *IEEE International Workshop on Signal Processing Systems*, Oct. 2017.
+
+[4] R. Rocher, D. Menard, P. Scalart, and O. Sentieys, "Analytical approach for numerical accuracy estimation of fixed-point systems based on smooth operations," *IEEE Transactions on Circuits and Systems*, pp. 2326-2339, Oct 2012.
+
+[5] K. Parashar, R. Rocher, D. Menard, and O. Sentieys, "Analytical approach for analyzing quantization noise effects on decision operators," in *IEEE International Conference on Acoustics, Speech and Signal Processing*, March 2010, pp. 1554-1557.
+
+[6] P. Roy, R. Ray, C. Wang, and W. F. Wong, "Asac: Automatic sensitivity analysis for approximate computing," in *SIGPLAN/SIGBED LCTES*. New York, NY, USA: ACM, 2014.
+
+[7] C. Rubio-González, C. Nguyen, H. D. Nguyen, J. Demmel, W. Kahan, K. Sen, D. H. Bailey, C. Iancu, and D. Hough, "Precimonious: Tuning assistant for floating-point precision," in *SC*, Nov 2013.
+
+[8] K. N. Parashar, D. Menard, and O. Sentieys, "Accelerated performance evaluation of fixed-point systems with un-smooth operations," *Transactions on Computer-Aided Design of Integrated Circuits and Systems*, pp. 599-612, April 2014.
+
+[9] G. Hamerly and J. Drake, *Accelerating Lloyd's Algorithm for k-Means Clustering*. Springer International Publishing, 2015.
+
+[10] J. Silva, E. Faria, R. Barros, E. Hruschka, A. de Carvalho, and J. Gama, "Data stream clustering: A survey," 2014.
+
+[11] P. Seidel, "A parametric error analysis of goldschmidt's square-root algorithm," in *ACSSC*, Nov 2015, pp. 727-731.
+
+[12] E. Vasilakis, "An instruction level energy characterization of arm processors," https://www.ics.forth.gr/carv/greenvm/files/tr450.pdf, CARV Laboratory ICS FORTH, Tech. Rep., 2015, 7/8/2018.
+
+[13] N. J. Higham, *Accuracy and Stability of Numerical Algorithms*. Society for Industrial and Applied Mathematics, 2002.
+
+[14] A. K. Jain, M. N. Murty, and P. J. Flynn, "Data clustering: A review," *ACM Comput. Surv.*, Sep. 1999.
+
+[15] P. Sinha and R. Russell, "A perceptually based comparison of image similarity metrics," *Perception*, 2011.
+
+[16] M. Leeser, J. Theiler, M. Estlick, and J. J. Szymanski, "Design tradeoffs in a hardware implementation of the k-means clustering algorithm," in *IEEE SAM SP Workshop*, 2000.
+
+[17] L. Fousse, G. Hanrot, V. Lefèvre, P. Pélissier, and P. Zimmermann, "Mpfr: A multiple-precision binary floating-point library with correct rounding," *ACM Trans. Math. Softw.*, no. 2, Jun. 2007.
+
+[18] A. Yazdanbakhsh, D. Mahajan, P. Lotfi-Kamran, and H. Esmaeilzadeh, "Axbench: A benchmark suite for approximate computing across the system stack," 2016.
+
+[19] "Uncompressed rgb images," https://bitbucket.org/act-lab/axbench/src, accessed: 06/08/2018.

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+ON EXISTENCE AND UNIQUENESS OF ASYMPTOTIC N-SOLITON-LIKE
+SOLUTIONS OF THE NONLINEAR KLEIN-GORDON EQUATION
+
+XAVIER FRIEDERICH
+
+**ABSTRACT.** We are interested in solutions of the nonlinear Klein-Gordon equation (NLKG) in $\mathbb{R}^{1+d}$, $d \ge 1$, which behave as a soliton or a sum of solitons in large time. In the spirit of other articles focusing on the supercritical generalized Korteweg-de Vries equations and on the nonlinear Schrödinger equations, we obtain an *N*-parameter family of solutions of (NLKG) which converges exponentially fast to a sum of *N* given (unstable) solitons. For *N* = 1, this family completely describes the set of solutions converging to the soliton considered; for *N* ≥ 2, we prove uniqueness in a class with explicit algebraic rate of convergence.
+
+# 1. INTRODUCTION
+
+**1.1. Setting of the problem.** We consider the following nonlinear Klein-Gordon equation
+
+$$
+\text{(NLKG)} \qquad \partial_t^2 u = \Delta u - u + f(u),
+$$
+
+where $u$ is a real-valued function of $(t,x) \in \mathbb{R} \times \mathbb{R}^d$ and $f$ is a $\mathscr{C}^1$ real-valued function on $\mathbb{R}$. This
+equation classically rewrites as the following first order system in time:
+
+$$
+\text{(NLKG')} \qquad \partial_t U = \begin{pmatrix} 0 & I_d \\ \Delta - I_d & 0 \end{pmatrix} U + \begin{pmatrix} 0 \\ f(u) \end{pmatrix},
+$$
+
+where $U$ is the two-vector $\begin{pmatrix} u \\ \partial_t u \end{pmatrix}$.
+
+Let us denote by $F$ the unique primitive of $f$ on $\mathbb{R}$ which vanishes in $0$. We make the following assumptions:
+
+• if $d=1$,
+
+(H1) $f$ is odd and $f'(0) = 0$.
+
+(H2) There exists $r > 0$ such that $F(r) > \frac{1}{2}r^2$.
+
+• if $d \ge 2$,
+
+(H'1) $f$ is a pure $H^1$-subcritical nonlinearity $r \mapsto \lambda|r|^{p-1}r$, with $\lambda > 0$ and $p > 1$ if
+$d = 2$ and $p \in (1, \frac{d+2}{d-2})$ if $d \ge 3$.
+
+Assumption (H1) for $d=1$ or assumption (H'1) for $d \ge 2$ on the nonlinearity $f$ ensures that
+the Cauchy problem is locally well-posed in the energy space $H^1(\mathbb{R}^d) \times L^2(\mathbb{R}^d)$ [13, 24]. It is even
+globally well-posed if one assumes further sufficient smallness on the initial condition.
+
+Recall also that the following quantities are conserved for $H^1 \times L^2$-solutions $(u, \partial_t u)$ of (NLKG'):
+
+• the energy $\frac{1}{2} \int_{\mathbb{R}^d} \{((\partial_t u)^2 + |\nabla u|^2 + u^2 - 2F(u))\} (t,x) dx$
+
+• the momentum $\int_{\mathbb{R}^d} \{\partial_t u \nabla u\} (t,x) dx.$

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+Similarly there exists $C' > 0$ such that for $t$ large
+
+$$ (3.2) \quad \left\| \Phi_{A'_1, \dots, A'_N}(t) - \Phi_{A'_1, \dots, A'_{i_0-1}}(t) - \sum_{j=i_0}^{N} A'_j e^{-e_j t} Y_{+,j}(t) \right\|_{H^1 \times L^2} \le C' \sum_{j=i_0}^{N} e^{-(e_j+\sigma)t}. $$
+
+Using that $\Phi_{A_1, \dots, A_N}(t) = \Phi_{A'_1, \dots, A'_N}(t)$ and $\Phi_{A_1, \dots, A_{i_0-1}}(t) = \Phi_{A'_1, \dots, A'_{i_0-1}}(t)$, we deduce from (3.1) and (3.2) that for all $t$ sufficiently large
+
+$$ e^{-e_{i_0}t} |A_{i_0} - A'_{i_0}| \le C e^{-(e_{i_0}+\sigma)t}. $$
+
+Hence, letting $t \to +\infty$, we obtain $A_{i_0} - A'_{i_0} = 0$, which leads to a contradiction. This ends the proof of Theorem 1.2.
+
+**3.1. Compactness argument assuming uniform estimate.** The goal of this subsection is to explain how to prove Proposition 3.1; for this, we follow the strategy of Combet [4] and Côte and Muñoz [9], both inspired from pioneering work by Martel [19] and Côte, Martel and Merle [8]. One key ingredient in the construction is the obtainment of uniform estimates satisfied by a sequence of approximating solutions of (NLKG).
+
+We fix $j \in \{1, \dots, N\}$ and $A_j \in \mathbb{R}$. Let $(S_n)_n$ be an increasing sequence of time such that $S_n \to +\infty$. Let us consider $\mathbf{b}_n = (b_{n,k})_{j<k \le N} \in \mathbb{R}^{N-j}$ the generic term of a sequence of parameters to be determined, and let $u_n$ be the maximal solution of (NLKG) such that
+
+$$ (3.3) \qquad U_n(S_n) = \Phi(S_n) + A_j e^{-e_j S_n} Y_{+,j}(S_n) + \sum_{k>j} b_{n,k} Y_{+,k}(S_n), $$
+
+where $U_n := \begin{pmatrix} u_n \\ \partial_t u_n \end{pmatrix}$.
+
+Concerning $u_n$, we claim:
+
+**Proposition 3.2.** There exist $n_0 \ge 0$ and $t_0 > 0$ (independent of $n$) such that for each $n \ge n_0$, there exists $\mathbf{b}_n \in \mathbb{R}^{N-j}$ with $|\mathbf{b}_n| \le 2e^{-(e_j+2\sigma)t}$ and such that $U_n$ is defined on $[t_0, S_n]$ and satisfies
+
+$$ (3.4) \quad \forall t \in [t_0, S_n], \quad \|U_n(t) - \Phi(t) - A_j e^{-e_j t} Y_{+,j}(t)\|_{H^1 \times L^2} \le C e^{-(e_j+\sigma)t}. $$
+
+The $b_n$ take the role of modulation parameters and are to be determined (if indeed possible) so that $U_n$ fulfills (3.4), thus is a natural candidate in order to "approximate" the desired solution $U$ which is the object of Proposition 3.2.
+
+We postpone the proof of the previous statement at the next subsection; for the time being, let us assume that Proposition 3.2 is satisfied and let us show how it implies Proposition 3.1. In fact, the existence of $U$ is due to the continuity of the flow of (NLKG) for the weak $H^1 \times L^2$ topology. We explicit the construction of $U$ below, following the same strategy as [8, paragraph 2.2, step 2] or [9, section 4].
+
+*Proof of Proposition 3.1.* We observe that the sequence $(\|U_n(t_0)\|_{H^1\times L^2})_{n\in\mathbb{N}}$ is bounded; thus there exist a subsequence of $(U_n(t_0))_{n\in\mathbb{N}}$, say $(U_{n_k}(t_0))_{k\in\mathbb{N}}$, and $U_0 \in H^1(\mathbb{R})\times L^2(\mathbb{R})$ such that $(U_{n_k}(t_0))_{k\in\mathbb{N}}$ converges to $U_0$ in the sense of the weak topology in $H^1(\mathbb{R}) \times L^2(\mathbb{R})$. Let us consider $U$, defined as the maximal solution of (NLKG) such that $U(t_0) = U_0$.
+
+Let $t \ge t_0$. For $k$ sufficiently large, $S_{n_k} \ge t$ and thus $U_{n_k}$ is defined on $[t_0, t]$. By a standard result (we refer to [9, Lemma 10] and [28, Theorem 1.2]), $U$ is defined on $[t_0, t]$ and $(U_{n_k}(t))_{k}$ converges weakly to $U(t)$ in $H^1(\mathbb{R}) \times L^2(\mathbb{R})$.
+
+Moreover, by property of the weak limit,
+
+$$ 
+\begin{align*}
+& \|U(t) - \Phi(t) - A_j e^{-e_j t} Y_{+,j}(t)\|_{H^1 \times L^2} \le \liminf_{k \to +\infty} \|U_{n_k}(t) - \Phi(t) - A_j e^{-e_j t} Y_{+,j}(t)\|_{H^1 \times L^2} \\
+& \le C_0 e^{-(e_j + \sigma)t}.
+\end{align*}
+\hspace*{\fill} □
+$$

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+Now, the remainder of Section 3 is devoted to the proof of Proposition 3.2.
+
+**3.2.** **Proof of Proposition 3.2.** For ease of reading, we will drop the index $n$ for the rest of this subsection (except for $S_n$), that is, we will write $U$ for $U_n$, $\mathfrak{b}$ for $\mathfrak{b}_n$, etc.
+
+Let us introduce the following variable (which depends on $n$)
+
+$$W(t) := U(t) - \Phi(t) - A_j e^{-e_j t} Y_{+,j}(t)$$
+
+and for all $k \in \{1, \dots, N\}$,
+
+$$\alpha_{\pm,k}(t) := \langle W(t), Z_{\pm,k}(t) \rangle$$
+
+(which depends on $\mathfrak{b}$ in particular by definition of $U = U_n$ (3.3)). We denote also $\alpha_-(t) := (\alpha_{-,k}(t))_{j<k \le N}$.
+
+**3.2.1. Modulated final data and strategy of the proof of Proposition 3.2.** We make the first step in order to determine the appropriate modulation parameter $\mathfrak{b}$. We obtain $\mathfrak{b}$ as the solution of a well-chosen equation; this is the object of the following
+
+**Lemma 3.3.** There exists $n_0 \ge 0$ such that for all $n \ge n_0$ and for all $\mathfrak{a} \in \mathbb{R}^{N-j}$, there exists a unique $\mathfrak{b} \in \mathbb{R}^{N-j}$ such that $\|\mathfrak{b}\| \le 2\|\mathfrak{a}\|$ and $\alpha_-(S_n) = \mathfrak{a}$.
+
+*Proof.* Let us consider the linear application
+
+$$\Psi: \quad \mathbb{R}^{N-j} \to \mathbb{R}^{N-j} \\ 
+\mathfrak{b} = (b_l)_{j<l \le N} \mapsto \left( \sum_{l>j} b_l \langle Y_{+,l}(S_n), Z_{-,l}(S_n) \rangle \right)_{j<k \le N}.$$
+
+Its matrix in the canonical basis of $\mathbb{R}^{N-j}$ has generic entry $\psi_{k,l} := \langle Y_{+,j+l}(S_n), Z_{-,j+k}(S_n) \rangle$ where $(k,l) \in \{1, \dots, N\}^2$.
+
+Since $\psi_{k,l} = 1$ if $k=l$ and $|\psi_{k,l}| \le C_0 e^{-\sigma S_n}$ for $k \ne l$, with $C_0 > 0$ independent of $n$, we have $\Psi = Id + M$ with $\|M\| \le \frac{1}{2}$ for large values of $n$. Thus $\Psi$ is invertible (for $n$ large) and $\|\Psi^{-1}\| \le 2$. We deduce the content of Lemma 3.3 by taking $n_0$ large enough and by considering, for a given $\mathfrak{a} \in \mathbb{R}^{N-j}$, the element $\mathfrak{b} := \Psi^{-1}(\mathfrak{a})$. $\square$
+
+Roughly speaking, Lemma 3.3 reflects that estimate (3.4) is to be proven by choosing a relevant vector $\mathfrak{a} = a_-(S_n)$.
+
+The reason why we determine $\mathfrak{b}$ according to the value of $\alpha_-(S_n)$ essentially comes from the directions $Z_{-,k}$, which yield "instability" in some sense (given Claim 3.7 below), and also from definition (3.5) below.
+
+At this stage, we notice that we already have:
+
+**Claim 3.4.** *We have:*
+
+(1) $\forall k \in \{1, \dots, N\}, \quad |\alpha_{+,k}(S_n)| \le C|\mathfrak{b}|e^{-2\sigma S_n}$.
+
+(2) $\forall k \in \{1, \dots, j\}, \quad |\alpha_{-,k}(S_n)| \le C|\mathfrak{b}|e^{-2\sigma S_n}$.
+
+(3) $\|W(S_n)\|_{H^1 \times L^2} \le C|\mathfrak{b}|.$
+
+Let $t_0 > 0$ independent of $n$ to be chosen later and $\mathfrak{a} \in B_{\mathbb{R}^{N-j}}(e^{-(e_j+2\sigma)S_n})$ to be determined. We consider the associated data $\mathfrak{b}$ given by Lemma 3.3 and $U$ defined in (3.3).
+
+Let us define
+
+$$T(\mathfrak{a}) := \inf\{T \ge t_0 | \forall t \in [T, S_n], \|W(t)\|_{H^1 \times L^2} \le e^{-(e_j+\sigma)t} \text{ and } e^{(e_j+2\sigma)t} \alpha_-(t) \in B_{\mathbb{R}^{N-j}}(1)\}.$$
+
+We observe that Proposition 3.2 holds if for all $n$, we can find $\mathfrak{a}$ such that $T(\mathfrak{a}) = t_0$. In the rest of the proof, our goal is thus to prove the existence of such an element $\mathfrak{a}$.
+
+To this end, we will first of all improve the estimate on $\|W(t)\|_{H^1 \times L^2}$ which falls within the definition of $T(\mathfrak{a})$. This is the object of the following subsection. Then, we will only need to care about the second condition, which implies a control of $\alpha_-(t)$; this is done in subsection 3.2.3.

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+**3.2.2. Improvement of the estimate on $||W||_{H^1 \times L^2}$**. For notation purposes and ease of reading, we sometimes omit the index $n$ and also write $O(G(t))$ in order to refer to a function $g$ which a priori depends on $n$ and such that there exists $C \ge 0$ such that for all $n$ large and for all $t \in [t_n^*, S_n]$, $|g(t)| \le C|G(t)|$.
+
+**Lemma 3.5.** There exists $K_0 > 0$ such that for all $t \in [T(a), S_n]$,
+
+$$ ||W(t)||_{H^1 \times L^2} \le \frac{K_0}{t^{1/4}} e^{-(e_j + \sigma)t}. $$
+
+The whole subsection consists of the proof of this lemma.
+
+*Step 1: Estimates on $\alpha_{\pm,k}$. Let us begin with the computation of the time derivative of W.*
+
+**Claim 3.6.** We have for all $k \in \{1, \dots, N\}$,
+
+$$ (3.6) \quad \partial_t W = \begin{pmatrix} 0 & Id \\ \partial_x^2 - Id + f'(\varphi) & 0 \end{pmatrix} W + A_j e^{-e_j t} \left[ \begin{pmatrix} \beta_j \partial_x & Id \\ \partial_x^2 - Id + f'(Q_k) & \beta_j \partial_x \end{pmatrix} Y_{+,j} + e_j Y_{+,j} \right] \\ \qquad + A_j e^{-e_j t} \begin{pmatrix} 0 & 0 \\ f'(\phi) - f'(Q_k) & 0 \end{pmatrix} Y_{+,j} + \begin{pmatrix} 0 \\ g \end{pmatrix}, $$
+
+where $g := f(u) - f(\varphi) - f'(\varphi)(u - \varphi)$ satisfies
+
+$$ ||g(t)||_{L^{\infty}} = O(||u - \varphi||_{H^1}^2). $$
+
+*Proof.* Claim 3.6 follows from the fact that both $U$ and $\Phi$ satisfy (NLKG') and is also a consequence of the following Taylor inequality ($f$ is $\mathscr{C}^2$)
+
+$$ |f(u) - f(\varphi) - f'(\varphi)(u - \varphi)|(t) \le C ||u(t) - \varphi(t)||_{L^\infty}^2 $$
+
+and the Sobolev embedding $H^1(\mathbb{R}) \hookrightarrow L^\infty(\mathbb{R})$. $\square$
+
+Now, we are in a position to prove the following estimate on $\alpha_{\pm,k}$.
+
+**Claim 3.7.** For all $k \in \{1, \dots, N\}$ and for all $t \in [T(a), S_n]$, we have
+
+$$ (3.7) \quad \left| \frac{d}{dt} \alpha_{\pm,k}(t) \mp e_k \alpha_{\pm,k}(t) \right| \le C \left( e^{-4\sigma t} ||W(t)||_{H^1 \times L^2} + ||W(t)||_{H^1 \times L^2}^2 + e^{-(e_j+4\sigma)t} \right). $$
+
+*Proof.* Let $k \in \{1, \dots, N\}$. By means of (3.6) and since $\partial_t Z_{\pm,k} = -\beta_k \partial_x Z_{\pm,k}$, we compute
+
+$$ 
+\begin{aligned}
+\frac{d}{dt} \alpha_{\pm,k}(t) &= \langle \partial_t W, Z_{\pm,k} \rangle + \langle W, \partial_t Z_{\pm,k} \rangle \\
+&= \left\langle W, \begin{pmatrix} -\beta_k \partial_x & \partial_x^2 - Id + f'(\varphi) \\ Id & -\beta_k \partial_x \end{pmatrix} Z_{\pm,k} \right\rangle \\
+&\quad + A_j e^{-e_j t} \left\langle Y_{+,j}, \begin{pmatrix} -\beta_j \partial_x & \partial_x^2 - Id + f'(Q_k) \\ Id & -\beta_j \partial_x \end{pmatrix} Z_{\pm,k} \right\rangle \\
+&\quad + A_j e^{-e_j t} \left[ \begin{pmatrix} 0 & f'(Q_k) - f'(\varphi) \\ 0 & 0 \end{pmatrix} Z_{\pm,k} + e_j \langle Y_{+,j}, Z_{\pm,k} \rangle \right] \\
+&\quad + O(||U - \Phi||_{H^1 \times L^2}^2).
+\end{aligned}
+$$
+
+Let us notice first that
+
+$$ \left\langle W, \begin{pmatrix} -\beta_k \partial_x & \partial_x^2 - Id + f'(\varphi) \\ Id & -\beta_k \partial_x \end{pmatrix} Z_{\pm,k} \right\rangle = \left\langle W, \mathcal{H}_k Z_{\pm,k} \right\rangle + \left\langle W, \begin{pmatrix} 0 & f'(\varphi) - f'(Q_k) \\ 0 & 0 \end{pmatrix} Z_{\pm,k} \right\rangle. $$
+
+We have
+
+$$ \langle W, \mathcal{H}_k Z_{\pm,k} \rangle = \langle W, \pm e_k Z_{\pm,k} \rangle = \pm e_k \alpha_{\pm,k}. $$

samples/texts/7376768/page_13.md

@@ -0,0 +1,84 @@
+and
+
+$$
+\begin{align*}
+& \left| \left\langle W, \begin{pmatrix} 0 & f'(\varphi) - f'(Q_k) \\ 0 & 0 \end{pmatrix} Z_{\pm,k} \right\rangle \right| \\
+& \le \|W\|_{H^1 \times L^2} \| (f'(\varphi) - f'(Q_k)) Z_{\pm,k} \|_{H^1 \times L^2} \\
+& \le C \left\| \varphi - \sum_{i=1}^N Q_i \right\|_{L^\infty} \|W\|_{H^1 \times L^2} + C \|W\|_{H^1 \times L^2} \sum_{i \ne k} \|Q_i Z_{\pm,k}\|_{H^1 \times L^2} \\
+& \le C e^{-4\sigma t} \|W\|_{H^1 \times L^2}.
+\end{align*}
+$$
+
+Similarly, we have
+
+$$
+\left| \left\langle Y_{+,j}, \begin{pmatrix} 0 & f'(Q_k) - f'(\varphi) \\ 0 & 0 \end{pmatrix} Z_{\pm,k} \right\rangle \right| \le C e^{-4\sigma t}
+$$
+
+and
+
+$$
+\begin{align*}
+\left\langle Y_{+,j}, \begin{pmatrix} -\beta_j \partial_x & \partial_x^2 - Id + f'(Q_k) \\ Id & -\beta_j \partial_x \end{pmatrix} Z_{\pm,k} \right\rangle &= \langle Y_{+,j}, \mathcal{H}_k Z_{\pm,k} \rangle + (\beta_k - \beta_j) \langle Y_{+,j}, \partial_x Z_{\pm,k} \rangle \\
+&= \pm \langle Y_{+,j}, e_k Z_{\pm,k} \rangle + O(e^{-4\sigma t}).
+\end{align*}
+$$
+
+Indeed, we notice that
+
+$$
+(\beta_k - \beta_j) \langle Y_{+,j}, \partial_x Z_{\pm,k} \rangle = \begin{cases} 0 & \text{if } k = j \\ O(e^{-4\sigma t}) & \text{if } k \neq j. \end{cases}
+$$
+
+Hence, we obtain
+
+$$
+(3.8) \quad \frac{d}{dt} \alpha_{\pm,k}(t) = \pm e_k \alpha_{\pm,k} + O(e^{-4\sigma t} \|W\|_{H^1 \times L^2}) \\
+\phantom{(3.8)\quad} + A_j e^{-e_j t} [\pm \langle Y_{+,j}, e_k Z_{\pm,k} \rangle + e_j \langle Y_{+,j}, Z_{\pm,k} + O(e^{-4\sigma t} + \|U-\Phi\|^2_{H^1\times L^2}) ].
+$$
+
+Now, we observe that
+
+$$
+\pm e_k \langle Y_{+,j}, Z_{\pm,k} \rangle + e_j \langle Y_{+,j}, Z_{\pm,k} \rangle = O(e^{-4\sigma t}).
+$$
+
+This is clear if $k \neq j$ and for $k = j$, we have
+
+$$
+\pm e_j \langle Y_{+,j}, Z_{\pm,j} \rangle + e_j \langle Y_{+,j}, Z_{\pm,j} \rangle = \begin{cases} 0+0=0 & \text{if } \pm = + \\ -e_j + e_j = 0 & \text{if } \pm = -; \end{cases}
+$$
+
+indeed, we recall from Proposition 2.1
+
+$$
+(Y_{+,j}, Z_{+,j}) = 0 \quad \text{and} \quad (Y_{+,j}, Z_{-,j}) = 1.
+$$
+
+In addition, we have by the well-known inequality $(a+b)^2 \le 2(a^2+b^2)$,
+
+$$
+\|U - \Phi\|_{H^1 \times L^2}^2 \le C (\|W\|_{H^1 \times L^2}^2 + e^{-2e_j t}).
+$$
+
+Considering that $2e_j \ge e_j + 4\sigma$, we have thus finished the proof of the claim.
+
+**Step 2: Control of the stable directions.**
+
+**Claim 3.8.** We have for all $k \in \{1, \dots, N\}$, for all $t \in [T(a), S_n]$,
+
+$$
+(3.9) \qquad |α_{+,k}(t)| ≤ Ce^{-(e_j+4σ)t}.
+$$
+
+*Proof.* Due to Claim 3.7 and (3.5), we obtain
+
+$$
+\left| \frac{d}{dt} \alpha_{+,k}(t) - e_k \alpha_{+,k}(t) \right| \le C e^{-(e_j+4\sigma)t},
+$$
+
+that is, for all $t \in [T(a), S_n]$,
+
+$$
+|(e^{-e_k t} \alpha_{+,k}(t))'| \le C e^{-(e_j+e_k+4\sigma)t}.
+$$

samples/texts/7376768/page_14.md

@@ -0,0 +1,72 @@
+Integrating, we deduce that for all $t \in [T(\mathfrak{a}), S_n]$,
+
+$$|e^{-e_k S_n} \alpha_{+,k}(S_n) - e^{-e_k t} \alpha_{+,k}(t)| \le Ce^{-(e_j+e_k+4\sigma)t}.$$
+
+Thus,
+
+$$|\alpha_{+,k}(t)| \le |\alpha_{+,k}(S_n)| + C e^{-(e_j+4\sigma)t}.$$
+
+From Claim 3.4 and Lemma 3.3, we have
+
+$$
+\begin{align*}
+|\alpha_{+,k}(S_n)| &\le Ce^{-2\sigma S_n} |b| \\
+&\le Ce^{-2\sigma S_n} e^{-(e_j+2\sigma)S_n} \\
+&\le Ce^{-(e_j+4\sigma)t}.
+\end{align*}
+$$
+
+Consequently, Claim 3.8 indeed holds.
+
+**Step 3: Control of the unstable directions for $k \le j$.**
+
+**Claim 3.9.** We have for all $k \in \{1, \dots, j\}$, for all $t \in [T(\mathfrak{a}), S_n]$,
+
+$$ (3.10) \qquad |\alpha_{-,k}(t)| \le Ce^{-(e_j+4\sigma)t}. $$
+
+*Proof.* As in the preceding step, we have for all $k \in \{1, \dots, N\}$ and $t \in [T(\mathfrak{a}), S_n]$,
+
+$$ (3.11) \qquad \left| \frac{d}{dt} \alpha_{-,k}(t) + e_k \alpha_{-,k}(t) \right| \le Ce^{-(e_j+4\sigma)t}, $$
+
+which writes also
+
+$$ |(e^{e_k t} \alpha_{-,k}(t))'| \le Ce^{(e_k - e_j + 4\sigma)t}. $$
+
+For $k \le j$, we have $e_k \le e_j$, and so by integration, we obtain
+
+$$ |\alpha_{-,k}(t)| \le e^{e_k(S_n-t)} |\alpha_{-,k}(S_n)| + Ce^{-(e_j+4\sigma)t}. $$
+
+But again from Claim 3.4 and Lemma 3.3, we infer
+
+$$
+\begin{align*}
+e^{e_k(S_n-t)} |\alpha_{-,k}(S_n)| &\le Ce^{e_k(S_n-t)} e^{-2\gamma S_n} e^{-(e_j+2\sigma)S_n} \\
+&\le Ce^{(S_n-t)(e_k-e_j)} e^{-e_j t} e^{-4\sigma S_n} \\
+&\le Ce^{-(e_j+4\sigma)t}.
+\end{align*}
+$$
+
+Thus
+
+$$ \forall k \in \{1, \dots, j\}, \quad \forall t \in [T(\mathfrak{a}), S_n], \qquad |\alpha_{-,k}(t)| \le Ce^{-(e_j+4\sigma)t}. $$
+
+**Step 4: Control of a Lyapunov functional satisfying a coercivity property.** Let us consider
+
+$$
+\psi : \mathbb{R} \to \mathbb{R} \\
+x \mapsto \frac{2}{\pi} \operatorname{Arctan}(e^{-x}).
+$$
+
+We define for all $k = 1, \dots, N-1$,
+
+$$ \psi_k(t, x) := \psi\left(\frac{1}{\sqrt{t}}\left(x - \frac{\beta_{\eta(k)} + \beta_{\eta(k+1)}}{2}t - \frac{x_{\eta(k)} + x_{\eta(k+1)}}{2}\right)\right), $$
+
+and then
+
+$$
+\phi_1(t) = \psi_1(t) \\
+\phi_k(t) = \psi_k(t) - \psi_{k-1}(t) \text{ for all } k = 2, \dots, N-1, \\
+\phi_N(t) = 1 - \psi_{N-1}(t).
+$$
+
+Recall that the permutation $\eta$ has been chosen so that $-1 < \beta_{\eta(1)} < \dots < \beta_{\eta(N)} < 1$.

samples/texts/7376768/page_15.md

@@ -0,0 +1,60 @@
+Now, let us introduce for all $k \in \{1, \dots, N\}$
+
+$$ \mathcal{F}_{W,k}(t) = \int_{\mathbb{R}} (w_1^2 + (\partial_x w_1)^2 + w_2^2 - f'(Q_{\eta(k)})w_1^2 + 2\beta_{\eta(k)}\partial_x w_1 w_2) \phi_k \, dx, $$
+
+and
+
+$$ \mathcal{F}_W(t) := \sum_{k=1}^{N} \mathcal{F}_{W,k}(t). $$
+
+By means of Proposition 2.2 and a usual localization argument [21, 22], we obtain that $\mathcal{F}_W$ is coercive on a subspace of $H^1 \times L^2$ of finite codimension. More precisely, there exists $\mu > 0$ such that
+
+$$ (3.12) \quad \mathcal{F}_W(t) \geq \mu \|W(t)\|_{H^1 \times L^2}^2 - \frac{1}{\mu} \sum_{k=1}^{N} \left( \langle W, \partial_x R_k \rangle^2 + \langle W, Z_{+,k} \rangle^2 + \langle W, Z_{-,k} \rangle^2 \right). $$
+
+We state the following control about the derivative of $\mathcal{F}_W$:
+
+**Claim 3.10.** For $t_0$ large and for all $t \in [T(a), S_n]$,
+
+$$ (3.13) \qquad \left| \frac{d}{dt} \mathcal{F}_W(t) \right| \le \frac{C}{\sqrt{t}} \|W\|_{H^1 \times L^2}^2. $$
+
+*Proof.* Let us rewrite $\mathcal{F}_{W,k}$ differently, using the notations developed in the introduction. Relying on integrations by parts, our computations lead to:
+
+$$ \langle (H_{\eta(k)} W) \phi_k, W \rangle = \mathcal{F}_{W,k}(t) - \frac{1}{2} \int_{\mathbb{R}} w_1^2 \partial_x^2 \phi_k \, dx + \beta_{\eta(k)} \int_{\mathbb{R}} w_1 w_2 \partial_x \phi_k \, dx. $$
+
+Thus
+
+$$ (3.14) \qquad \mathcal{F}_{W,k}(t) = \langle (H_{\eta(k)} W)\phi_k, W\rangle + O\left(\frac{1}{\sqrt{t}}\|W\|_{H^1\times L^2}^2\right). $$
+
+We immediately have
+
+$$ \frac{d}{dt} \langle (H_{\eta(k)} W) \phi_k, W \rangle = \langle (H_{\eta(k)} W) \phi_k, \partial_t W \rangle + \langle \partial_t (H_{\eta(k)} W) \phi_k, W \rangle + \langle (H_{\eta(k)} W) \partial_t \phi_k, W \rangle. $$
+
+Besides
+
+$$ 
+\begin{align*}
+& \langle \partial_t (H_{\eta(k)} W) \phi_k, W \rangle + \langle (H_{\eta(k)} W) \partial_t \phi_k, W \rangle \\
+&= \langle (H_{\eta(k)} \partial_t W) \phi_k, W \rangle + \beta_{\eta(k)} \int_{\mathbb{R}} \partial_x Q_{\eta(k)} f''(Q_{\eta(k)}) w_1^2 \phi_k \, dx + O\left(\frac{1}{\sqrt{t}} \|W\|_{H^1 \times L^2}^2\right) \\
+&= \langle H_{\eta(k)} (\partial_t W), W \phi_k \rangle + \beta_{\eta(k)} \int_{\mathbb{R}} \partial_x Q_{\eta(k)} f''(Q_{\eta(k)}) w_1^2 \phi_k \, dx + O\left(\frac{1}{\sqrt{t}} \|W\|_{H^1 \times L^2}^2\right).
+\end{align*}
+$$
+
+Since $H_{\eta(k)}$ is a self-adjoint operator, we have
+
+$$ \langle H_{\eta(k)} (\partial_t W), W \phi_k \rangle = \langle \partial_t W, H_{\eta(k)} (W \phi_k) \rangle. $$
+
+By a straightforward calculation, we have moreover
+
+$$ \langle H_{\eta(k)}(W\phi_k), \partial_t W \rangle = (\langle H_{\eta(k)}(W)\phi_k, \partial_t W \rangle + O(\frac{1}{\sqrt{t}} \|W\|_{H^1\times L^2})). $$
+
+At this stage, we thus obtain
+
+$$ 
+\begin{align*}
+& \frac{d}{dt} \langle (H_{\eta(k)} W) \phi_k, W \rangle = 2\langle (H_{\eta(k)} W) \phi_k, \partial_t W \rangle \\
+& \phantom{{}= 2\langle (H_{\eta(k)} W) } + \beta_{\eta(k)} \int_{\mathbb{R}} \partial_x Q_{\eta(k)} f''(Q_{\eta(k)}) w_1^2 \phi_k dx + O\left(\frac{1}{\sqrt{t}} \|W\|_{H^1\times L^2}\right).
+\end{align*}
+$$
+
+Now, by (3.6), we write
+
+$$ (\langle H_{\eta(k)} W )\phi_k , \partial_t W\rangle = I_1 + I_2 + I_3 $$

samples/texts/7376768/page_16.md

@@ -0,0 +1,54 @@
+where
+
+$$
+I_1 := \left\langle \begin{pmatrix} T_{\eta(k)} & 0 \\ 0 & Id \end{pmatrix} W \phi_k, \begin{pmatrix} 0 & 0 \\ \partial_x^2 - Id + f'(\varphi) & 0 \end{pmatrix} \right\rangle W
+$$
+
+by denoting $T_i = -\partial_x^2 + Id - f'(Q_i)$ for all $i = 1, \dots, N$,
+
+$$
+I_2 := \beta_{\eta(k)} \left\langle \begin{pmatrix} 0 & -\partial_x \\ \partial_x & 0 \end{pmatrix} W \phi_k, \begin{pmatrix} 0 & Id \\ -T_{\eta(k)} + f'(\varphi) - f'(Q_{\eta(k)}) & 0 \end{pmatrix} \right\rangle_W
+$$
+
+and
+
+$$
+I_3 := A_j e^{-e_j t} \left\langle (H_{\eta(k)} W) \phi_k, \begin{pmatrix} \beta_j \partial_x & Id \\ \partial_x^2 - Id + f'(Q_j) & \beta_j \partial_x \end{pmatrix} Y_{+,j} \right\rangle \\
+\qquad + \left\langle (H_{\eta(k)} W) \phi_k, e_j Y_{+,j} + \begin{pmatrix} 0 & 0 \\ f'(\varphi) - f'(Q_j) & 0 \end{pmatrix} Y_{+,j} \right\rangle.
+$$
+
+Let us deal with $I_1$: we observe that
+
+$$
+\begin{align*}
+I_1 &= \left\langle \begin{pmatrix} T_{\eta(k)} & 0 \\ 0 & Id \end{pmatrix} W \phi_k, \begin{pmatrix} 0 & 0 \\ -T_{\eta(k)} + f'(\varphi) - f'(Q_{\eta(k)}) & Id \end{pmatrix} W \right\rangle \\
+&= \left\langle W, \begin{pmatrix} T_{\eta(k)} & 0 \\ 0 & Id \end{pmatrix} \begin{pmatrix} 0 & Id \\ -T_{\eta(k)} & 0 \end{pmatrix} W \phi_k \right\rangle + O(e^{-4\sigma t} \|W\|_{H^1\times L^2}^2) \\
+&= \left\langle W, \begin{pmatrix} 0 & T_{\eta(k)} \\ -T_{\eta(k)} & 0 \end{pmatrix} W \phi_k \right\rangle + O(e^{-4\sigma t} \|W\|_{H^1\times L^2}^2) \\
+&= -\int_{\mathbb{R}} w_1 \partial_x^2 w_2 \phi_k dx + \int_{\mathbb{R}} w_2 \partial_x^2 w_1 \phi_k dx + O\left(\frac{1}{\sqrt{t}} \|W\|_{H^1\times L^2}^2\right) \\
+&= O\left(\frac{1}{\sqrt{t}} \|W\|_{H^1\times L^2}^2\right).
+\end{align*}
+$$
+
+We have
+
+$$
+I_2 = \beta_{\eta(k)} \left\langle \begin{pmatrix} 0 & -\partial_x \\ \partial_x & 0 \end{pmatrix} W \phi_k, \begin{pmatrix} 0 & Id \\ -T_{\eta(k)} & 0 \end{pmatrix} W \right\rangle + O\left(e^{-(e_j+4\sigma)t} \|W\|_{H^1\times L^2}\right) \\
+= -\beta_{\eta(k)} \int_{\mathbb{R}} \partial_x w_2 w_2 \phi_k dx + \beta_{\eta(k)} \int_{\mathbb{R}} \partial_x w_1 (\partial_x^2 w_1 - w_1 + f'(Q_{\eta(k)})w_1) \phi_k dx \\
++ O\left(e^{-(e_j+4\sigma)t} \|W\|_{H^1\times L^2}\right) \\
+= -\frac{\beta_{\eta(k)}}{2} \int_{\mathbb{R}} w_1^2 \partial_x Q_{\beta_{\eta(k)}} f''(Q_{\beta_{\eta(k)}}) \phi_k dx + O\left(\frac{1}{\sqrt{t}} \|W\|_{H^1\times L^2}^2 + e^{-(e_j+4\sigma)t} \|W\|_{H^1\times L^2}\right).
+$$
+
+In addition, we have
+
+$$
+I_3 = A_j e^{-e_j t} \langle (H_{\eta(k)} W) \phi_k, JZ_{+,j} + e_j Y_{+,j} \rangle + O \left( e^{-(4\sigma+e_j)t} \|W\|_{H^1\times L^2} \right) \\
+= A_j e^{-e_j t} \left( \langle W\phi_k, -\mathcal{H}_{\eta(k)} Z_{+,j} \rangle + e_j \langle W\phi_k, H_{\eta(k)} Y_{+,k} \rangle \right) \\
++ O \left( \frac{1}{\sqrt{t}} \|W\|_{H^1\times L^2}^2 + e^{-(4\sigma+e_j)t} \|W\|_{H^1\times L^2} \right) \\
+= O \left( \frac{1}{\sqrt{t}} \|W\|_{H^1\times L^2}^2 + e^{-(4\sigma+e_j)t} \|W\|_{H^1\times L^2} \right).
+$$
+
+Note that the last line of the previous equality is a consequence of the following observation: if $\eta(k) = j$, we have $\mathcal{H}_j Z_{+,j} = e_j Z_{+,j}$ and $H_j Y_{+,j} = Z_{+,j}$ so that
+
+$$
+A_j e^{-e_j t} (\langle W \phi_k, -\mathcal{H}_{\eta(k)} Z_{+,j} \rangle + e_j \langle W \phi_k, H_{\eta(k)} Y_{+,j} \rangle) = 0.
+$$

samples/texts/7376768/page_17.md

@@ -0,0 +1,49 @@
+If $\eta(k) \neq j$, we have
+
+$$A_j e^{-e_j t} (\langle W\phi_k, -\mathcal{H}_{\eta(k)}Z_{+,j} \rangle + e_j \langle W\phi_k, H_{\eta(k)}Y_{+,j} \rangle) = O(e^{-(e_j+4\sigma)t} \|W\|_{H^{1\times L^2}}).$$
+
+Gathering the preceding computations yields
+
+$$\left|\frac{d}{dt}\mathcal{F}_{W,k}(t)\right| \le C \left(\frac{1}{\sqrt{t}}\|\mathcal{W}\|_{H^{1\times L^2}}^2 + e^{-(e_j+4\sigma)t}\|\mathcal{W}\|_{H^{1\times L^2}}^2\right),$$
+
+hence the expected claim, by summing on $k$. $\square$
+
+**Step 5: Control of the directions $\partial_x R_k$.** To obtain a control of the scalar products $\langle W, \partial_x R_k \rangle$ which is more precise than the a priori control by $\|\tilde{W}\|_{H^{1\times L^2}}$, let us introduce the following modulated variable $\tilde{W}$:
+
+$$ (3.15) \qquad \tilde{W}(t) = W(t) + \sum_{k=1}^{N} a_k(t) \partial_x R_k(t), $$
+
+where $a_k(t) \in \mathbb{R}, k = 1, \dots, N$ are chosen so that for all $l = 1, \dots, N$, $\langle \tilde{W}(t), \partial_x R_l(t) \rangle = 0$. Existence and uniqueness of the family $(a_k(t))_{k \in \{1, \dots, N\}}$ are justified by the fact that the (interaction) $N \times N$-matrix with generic entry $\langle \partial_x R_k(t), \partial_x R_l(t) \rangle$ is invertible for $t$ large enough.
+
+Notice that
+
+$$ (3.16) \qquad |a_k(t)| \le C \|W(t)\|_{H^{1\times L^2}} \le Ce^{-(e_j+\sigma)t} $$
+
+The functional $\mathcal{F}_{\tilde{W}}(t)$, defined as $\mathcal{F}_W(t)$ by changing $W$ in $\tilde{W}$, satisfies the following coercivity property:
+
+$$ (3.17) \qquad \|\tilde{W}\|_{H^{1\times L^2}}^2 \le C \left( \mathcal{F}_{\tilde{W}}(t) + \sum_{k=1}^{N} \left( \langle \tilde{W}, Z_{+,k} \rangle^2 + \langle \tilde{W}, Z_{-,k} \rangle^2 \right) \right). $$
+
+We have
+
+$$ \mathcal{F}_{\tilde{W}}(t) \le \mathcal{F}_W(t) + O\left(e^{-4\sigma t} \|W\|_{H^{1\times L^2}}^2\right) $$
+
+and we have moreover by Proposition 2.3 and (3.15).
+
+$$ \langle \tilde{W}, Z_{\pm,k} \rangle^2 \le \alpha_{\pm,k}^2 + e^{-2(e_j+5\sigma)t}. $$
+
+**Claim 3.11 (Estimate on $\tilde{W}$).** We have
+
+$$ \forall t \in [T(a), S_n], \qquad \|\tilde{W}(t)\|_{H^{1\times L^2}}^2 \le \frac{1}{\sqrt{t}} e^{-(2e_j+2\sigma)t}. $$
+
+*Proof.* Let $t$ belong to $[T(a), S_n]$. We obtain by (3.17) and by integration of (3.13) on $[t, +\infty)$ (which is indeed possible by definition of $T(a)$) that
+
+$$ \|\tilde{W}(t)\|_{H^{1\times L^2}}^2 \le \frac{C}{\sqrt{t}} e^{-2(e_j+\sigma)t} + C \sum_{\pm,k} \alpha_{\pm,k}^2 + C e^{-2(e_j+4\sigma)t}. $$
+
+Using the estimate on $\alpha_{\pm,k}$ provided by the definition of $T(a)$ and Claim 3.8, we then infer:
+
+$$ \|\tilde{W}(t)\|_{H^{1\times L^2}}^2 \le \frac{C}{\sqrt{t}} e^{-2(e_j+\sigma)t} + Ce^{-(2e_j+4\sigma)t}. $$
+
+This concludes the proof of the claim. $\square$
+
+**Claim 3.12 (Control of the modulation parameters).** We have for all $k = 1, \dots, N$,
+
+$$ \forall t \in [T(a), S_n], \quad |a_k(t)| \le \frac{C}{t^{4/3}} e^{-(e_j+\sigma)t}. $$

samples/texts/7376768/page_18.md

@@ -0,0 +1,80 @@
+*Proof.* By definition of the modulation parameters $a_k$, we have $\langle \tilde{W}, \partial_x R_k \rangle = 0$. Thus, we have by differentiation with respect to $t$:
+
+$$
+\langle \partial_t \tilde{W}, \partial_x R_k \rangle + \langle \tilde{W}, \partial_t \partial_x R_k \rangle = 0.
+$$
+
+By Proposition 2.3, we have for $l \neq k$,
+
+$$
+\langle \partial_x R_l, \partial_x R_k \rangle = O(e^{-4\sigma t})
+$$
+
+and for all $l$,
+
+$$
+\langle \partial_t \partial_x R_l, \partial_x R_k \rangle = O(e^{-4\sigma t}).
+$$
+
+We deduce that
+
+$$
+a'_{k}(t) \langle \partial_{x} R_{k}, \partial_{x} R_{k} \rangle + \langle \partial_{t} W, \partial_{x} R_{k} \rangle + \langle \tilde{W}, \partial_{t} \partial_{x} R_{k} \rangle = O \left( e^{-4\sigma t} \|W(t)\|_{H^{1} \times L^2} \right).
+$$
+
+We have in addition
+
+$$
+\left\langle W, \begin{pmatrix} 0 \\ Id \end{pmatrix} \partial_x^2 - Id + f'(\varphi) \begin{pmatrix} 0 \\ 0 \end{pmatrix} \partial_x R_k \right\rangle + \langle \tilde{W}, \partial_{t,x} R_k \rangle = O (\|\tilde{W}\|_{H^{1} \times L^2}) .
+$$
+
+What is more,
+
+$$
+|A_j e^{-e_j t} e_j \langle Y_{+,j}, \partial_x R_k \rangle| \le C e^{-(e_j+4\sigma)t},
+$$
+
+again by Proposition 2.3.
+
+Hence,
+
+$$
+\begin{align*}
+|a'_k(t)| &\le C \| \tilde{W}(t) \|_{H^{1} \times L^2} + C e^{-(e_j+4\sigma)t} \\
+&\le \frac{C}{t^{1/4}} e^{-(e_j+\sigma)t} + e^{-(e_j+3\sigma)t}.
+\end{align*}
+$$
+
+$\square$
+
+Now, gathering (3.15), Claim 3.11, and Claim 3.12, we immediately deduce the expected estimate of $\|W\|_{H^1\times L^2}$, which ends the proof of Lemma 3.5.
+
+**3.2.3. Control of the unstable directions for $k > j$ and end of the proof.** To control $\alpha_- = (\alpha_{-,k})_{j<k\le N}$ and eventually obtain the following statement, we resort to a classical topological argument, already set up in [4] and initially developed by Côte, Martel and Merle [8].
+
+**Lemma 3.13.** For $t_0$ large enough, there exists $\mathfrak{a} \in B_{\mathbb{R}^{N-j}}(e^{-(e_j+2\sigma)S_n})$ such that $T(\mathfrak{a}) = t_0$.
+
+The proof follows that of Combet [4]. We write it below for the sake of completeness.
+
+*Proof.* We first choose $t_0$ sufficiently large such that $\frac{K_0}{\sqrt{t_0}} \le \frac{1}{2}$. Then, we have by Lemma 3.5
+
+$$
+\forall t \in [T(\mathbf{a}), S_n], \quad \|W(t)\|_{H^1 \times L^2} \le \frac{1}{2} e^{-(e_j+\sigma)t}.
+$$
+
+Assume, for the sake of contradiction, that for all $\mathfrak{a} \in B_{\mathbb{R}^{N-j}}(e^{-(e_j+2\sigma)S_n})$, $T(\mathfrak{a}) > t_0$. As $\|W(T(\mathfrak{a}))\|_{H^1\times L^2} \le \frac{1}{2}e^{-(e_j+\sigma)T(\mathfrak{a})}$, by definition of $T(\mathfrak{a})$ and continuity of the flow, we have:
+
+$$
+|\alpha_-(T(\mathbf{a}))| = 1.
+$$
+
+(We recall that $\alpha_-(t) = (\alpha_{-,k}(t))_{j<k\le N}$.) In other words, the map
+
+$$
+\begin{array}{r@{\quad}c@{\quad}l@{\quad}\rightrightarrows}
+\mathcal{M}: & B_{\mathbb{R}^{N-j}}(e^{-(e_j+2\sigma)S_n}) & S_{\mathbb{R}^{N-j}}(e^{-(e_j+2\sigma)S_n}) \\
+& \mathfrak{a} & e^{-(e_j+2\sigma)(S_n-T(\mathfrak{a}))}\alpha_-(T(\mathfrak{a}))
+\end{array}
+$$
+
+is well-defined. Now, we aim at showing that $\mathcal{M}$ is continuous and that its restriction to $S_{\mathbb{R}^{N-j}}(e^{-(e_j+2\gamma)S_n})$
+is the identity.

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+Let $T \in [T_0, T(\mathbf{a})]$ be such that $W$ is defined on $[T, S_n]$ and, by continuity,
+
+$$\forall t \in [T, S_n], \quad \|W(t)\|_{H^{1} \times L^2} \le 1.$$
+
+We consider, for all $t \in [T, S_n]$:
+
+$$\mathcal{N}(t) := \mathcal{N}(\alpha_{-}(t)) = \|e^{(e_j+2\sigma)t}\alpha_{-}(t)\|^2.$$
+
+**Claim 3.14.** For $t_0$ large enough, and for all $t \in [T, S_n]$ such that $\mathcal{N}(t) = 1$, we have:
+
+$$\mathcal{N}'(t) \le -(e_{j+1} - e_j - 2\sigma).$$
+
+*Proof of Claim 3.14.* Let us start from estimate (3.11): for all $k \in \{j+1, \dots, N\}$, for all $t \in [T, S_n]$,
+
+$$\left|\frac{d}{dt}\alpha_{-,k} + e_k\alpha_{-,k}\right| \le Ce^{-(e_j+4\sigma)t}.$$
+
+Thus we obtain for all $k \in \{j+1, \dots, N\}$,
+
+$$\alpha_{-,k} \frac{d}{dt}\alpha_{-,k} + e_{j+1}\alpha_{-,k}^2 \le \alpha_{-,k} \frac{d}{dt}\alpha_{-,k} + e_k\alpha_{-,k}^2 \le Ce^{-(e_j+4\sigma)t}|\alpha_{-,k}|.$$
+
+Then, summing on $k \in \{j+1, \dots, N\}$ leads to
+
+$$(|\alpha_{-}(t)|^2)' + 2e_{j+1}|\alpha_{-}(t)|^2 \le Ce^{-(e_j+4\sigma)t}|\alpha_{-}(t)|.$$
+
+Therefore we can estimate:
+
+$$\begin{aligned} \mathcal{N}'(t) &= e^{2(e_j+2\sigma)t} \left[ 2(e_j + 2\sigma)|\alpha_{-}(t)|^2 + (|\alpha_{-}(t)|^2)' \right] \\ &\le e^{2(e_j+2\sigma)t} \left[ 2(e_j + 2\sigma)|\alpha_{-}(t)|^2 - 2e_{j+1}|\alpha_{-}(t)|^2 + Ce^{-(e_j+4\sigma)t}|\alpha_{-}(t)| \right]. \end{aligned}$$
+
+Hence we have for all $t \in [T, S_n]$,
+
+$$\mathcal{N}'(t) \le -\theta\mathcal{N}(t) + Ce^{e_j t}|\alpha_{-}(t)|,$$
+
+where $\theta = 2(e_{j+1} - e_j - 2\sigma) > 0$ by definition of $\sigma$. In particular, for all $\tau \in [T, S_n]$ satisfying $\mathcal{N}(\tau) = 1$, we have:
+
+$$\mathcal{N}'(\tau) \le -\theta + Ce^{e_j \tau} |\alpha_{-}(\tau)| \le -\theta + Ce^{e_j \tau} e^{-(e_j+2\sigma)\tau} \le -\theta + Ce^{-2\sigma t_0}.$$
+
+Now, we fix $t_0$ large enough such that $Ce^{-2\sigma t_0} \le \frac{\theta}{2}$. Thus for all $\tau \in [T, S_n]$ such that $\mathcal{N}(\tau) = 1$, we have
+
+$$\mathcal{N}'(\tau) \le -\frac{\theta}{2}.$$
+
+□
+
+Finally, we claim that $\mathbf{a} \mapsto T(\mathbf{a})$ is continuous. Indeed, let $\varepsilon > 0$. By definition of $T(\mathbf{a})$ and by Claim 3.14, there exists $\delta > 0$ such that for all $t \in [T(\mathbf{a}) + \varepsilon, S_n]$, $\mathcal{N}(t) < 1 - \delta$, and such that $\mathcal{N}(T(\mathbf{a}) - \varepsilon) > 1 + \delta$. But from continuity of the flow, there exists $\eta > 0$ such that for all $\tilde{\mathbf{a}}$ satisfying $\|\tilde{\mathbf{a}} - \mathbf{a}\| \le \eta$, we have
+
+$$\forall t \in [T(\mathbf{a}) - \varepsilon, S_n], \quad |\mathcal{N}(\tilde{\mathbf{a}}) - \mathcal{N}(\mathbf{a})| \le \frac{\delta}{2}.$$
+
+We finally deduce that
+
+$$T(\mathbf{a}) - \varepsilon \le T(\tilde{\mathbf{a}}) \le T(\mathbf{a}) + \varepsilon.$$
+
+Hence, $\mathbf{a} \mapsto T(\mathbf{a})$ is continuous.
+
+We then obtain that the map $\mathcal{M}$ is continuous. What is more, for $\mathbf{a} \in S_{\mathbb{R}^{N-j}}(e^{-(e_j+2\sigma)S_n})$, as $\mathcal{N}'(S_n) \le -(e_{j+1} - e_j - 2\sigma) < 0$, we then deduce by definition of $T(\mathbf{a})$ that $T(\mathbf{a}) = S_n$, and thus, $\mathcal{M}(\mathbf{a}) = \mathbf{a}$.
+
+The existence of such a map $\mathcal{M}$ contradicts Brouwer's fixed point theorem. Thus, we have finished proving Lemma 3.13.
+
+□

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+Moreover, the structure of the equation is left invariant under the action of $\mathbb{R} \times \mathbb{R}^d$ by (time and space) translation, and under the action of the Lorentz group $O(1, d)$ which consists of the linear automorphisms of $\mathbb{R}^{1+d}$ that preserve the quadratic form $(t, x_1, \dots, x_d) \mapsto t^2 - \sum_{i=1}^d x_i^2$. In other words, precising this latter action, for all $\beta \in \mathbb{R}^d$ with $\mathbb{R}^d$-euclidean norm $|\beta| < 1$ and $\gamma := \frac{1}{\sqrt{1-|\beta|^2}}$, $u$ is a solution of (NLKG) if and only if
+
+$$ (t, x) \mapsto u(\Lambda_{\beta}(t, x)) $$
+
+is still a solution to (NLKG), where $\Lambda_{\beta}$ is the linear transformation with matrix
+
+$$ \begin{pmatrix} \gamma & -\gamma\beta \\ -\gamma\beta^{\top} & I_d + \frac{\gamma-1}{|\beta|^2}\beta^{\top}\beta \end{pmatrix} $$
+
+in the canonical basis of $\mathbb{R}^{1+d}$. We observe in particular that
+
+$$ \Lambda_{\beta}(t,x) = (\gamma(t-\beta x), \gamma(x-\beta t)) $$
+
+if $d=1$. We refer to [9] for further details concerning the Lorentz transformations in all dimensions.
+
+It is well-known that (NLKG) admits a family of solitons indexed by two parameters: the velocity parameter $\beta \in \mathbb{R}^d$ with $|\beta| < 1$ and the translation parameter $x_0 \in \mathbb{R}^d$. Let $Q$ denote the unique (up to translation) positive $H^1$ solution of the following stationary elliptic problem, associated with (NLKG):
+
+$$ (1.1) \qquad \Delta Q - Q + f(Q) = 0 $$
+
+which we take as radial; for the record, existence of $Q$ follows from a standard result of Berestycki and Lions [2] due to (H2) or (H'1) and uniqueness has been proved in Kwong [17] (in the case where $f(u) = |u|^{p-1}u$ is the particular power nonlinearity) and in Serrin and Tang [27]. We recall that $Q$ and its partial derivatives up to order 3 decay exponentially. Then for all $\beta \in \mathbb{R}^d$ such that $|\beta| < 1$, for all $x_0 \in \mathbb{R}^d$, the boosted ground state
+
+$$ Q_{\beta,x_0} : (t,x) \mapsto Q(p r \circ \Lambda_{\beta}(t, x - x_0)), $$
+
+where $\gamma := \frac{1}{\sqrt{1-|\beta|^2}}$ and $pr$ is the canonical projection $\mathbb{R}^{1+d} \to \mathbb{R}^d$ on the last $d$ coordinates, is a solution of (NLKG) known as *soliton*. In the one-dimensional case, this soliton rewrites
+
+$$ Q_{\beta,x_0} : (t,x) \mapsto Q(x - \beta t - x_0). $$
+
+Soliton theory concerning (NLKG) has extensively been studied in many articles. One major result is linked to the classification of the solutions with energy near that of the ground state. Dynamics of the solutions $u$ of (NLKG) on the threshold energy $E(u) = E(Q)$ has been investigated in Duyckaerts and Merle [10]. More generally, classification of the solutions with energy less than a quantity slightly larger than the energy of the ground state has been done by Nakanishi and Schlag [25] and by Krieger, Nakanishi and Schlag [16].
+
+Let us also mention that solitons of (NLKG) are known to be orbitally unstable in $H^1(\mathbb{R}^d)$ by a general property by Grillakis, Shatah and Strauss [14].
+
+We further develop soliton analysis by exploring solutions which behave as a soliton or a sum of solitons as time goes to infinity.
+
+For all $\beta \in \mathbb{R}^d$ such that $|\beta| < 1$ and $x_0 \in \mathbb{R}^d$, let us denote
+
+$$ R_{\beta,x_0}(t,x) := \begin{pmatrix} Q_{\beta,x_0}(t,x) \\ \partial_t Q_{\beta,x_0}(t,x) \end{pmatrix} = \begin{pmatrix} Q_{\beta,x_0}(t,x) \\ -\beta \cdot \nabla Q_{\beta,x_0}(t,x) \end{pmatrix}. $$
+
+When $x_0 = 0$, we will write $R_{\beta}$ instead of $R_{\beta,0}$ for the sake of simplification.
+
+Drawing on the work by Grillakis, Shatah and Strauss [14, 15], Côte and Muñoz [9] have developed and proved spectral results adapted to the unstable dynamic around the (vector) soliton $R_{\beta}$. Essential

samples/texts/7376768/page_20.md

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+**4. CLASSIFICATION UNDER CONDITION OF THE MULTI-SOLITONS OF (NLKG)**
+
+Let $N \ge 2$ and $x_1, \dots, x_N, \beta_1, \dots, \beta_N$ be $2N$ parameters as in Theorem 1.2. Let $U$ be a solution of (NLKG) such that
+
+$$
+(4.1) \qquad \left\| U(t) - \sum_{i=1}^{N} R_{\beta_i}(t) \right\|_{H^1 \times L^2} = O \left( \frac{1}{t^\alpha} \right) \quad \text{as } t \to +\infty
+$$
+
+for some $\alpha > 3$.
+
+The goal of this section is to prove the existence of $A_1, \dots, A_N \in \mathbb{R}$ such that
+
+$$
+U = \Phi_{A_1, \dots, A_N}.
+$$
+
+Here again, we make the proof for $d = 1$.
+
+We denote by $\varphi$ a multi-soliton solution associated with these parameters, satisfying (2.2) and
+$\Phi := \begin{pmatrix} \varphi \\ \partial_t \varphi \end{pmatrix}$. Let us consider $Z := U - \Phi = \begin{pmatrix} z \\ \partial_t z \end{pmatrix}$. Obviously,
+
+$$
+\|Z(t)\|_{H^1 \times L^2} = O\left(\frac{1}{t^\alpha}\right), \quad \text{as } t \to +\infty.
+$$
+
+Our first objective is to improve this comparison, and namely to pass from the polynomial decay
+to an exponential one.
+
+**4.1. Exponential convergence to 0 at speed $e_1$ of $\|Z(t)\|_{H^1 \times L^2}$**
+
+**4.1.1. Introduction of a new variable by modulation.** In a standard way, we modulate the vari-
+able Z in order to obtain suitable orthogonality properties, making it possible to obtain crucial esti-
+mates when we apply the spectral theory available for (NLKG).
+
+**Lemma 4.1.** There exists $t_0 > 0$ and $\mathscr{C}^1$ functions $a_i : [t_0, +\infty) \to \mathbb{R}$ and $b_i : [t_0, +\infty) \to \mathbb{R}$ for all $i = 1, \dots, N$ such that, defining
+
+$$
+E := Z - \sum_{i=1}^{N} a_i \partial_x R_i - \sum_{i=1}^{N} b_i Y_{+,i},
+$$
+
+we have for all $i = 1, \dots, N$ and for all $t \ge t_0$:
+
+$$
+(4.2) \qquad \langle E(t), \partial_x R_i(t) \rangle = 0
+$$
+
+$$
+(4.3) \qquad \langle E(t), Z_{-,i}(t) \rangle = 0.
+$$
+
+Moreover, we have for all *i* = 1, ..., *N*:
+
+$$
+(4.4) \qquad a_i(t) = \frac{1}{\|\partial_x R_i\|^2} \langle Z(t), \partial_x R_i(t) \rangle + O(e^{-4\sigma t} \|Z\|_{H^1\times L^2})
+$$
+
+$$
+(4.5) \qquad b_i(t) = \langle Z(t), Z_{-,i}(t) \rangle + O\left(e^{-4\sigma t} \|Z\|_{H^1\times L^2}\right).
+$$
+
+Proof. This lemma follows from the consideration of the system with unknown variables $a_i$ and $b_i$ which is obtained by replacing $E$ by its definition in (4.2) and (4.3). See also [6] for similar considerations in the case of modulation for the nonlinear Schrödinger equations. □
+
+**4.1.2. Control of the Z<sub>+,i</sub> and Z<sub>-,i</sub> directions.** Define α<sub>±,i</sub> := ⟨Z, Z<sub>±,i</sub>⟩ for all i = 1, ..., N. We
+claim:
+
+**Lemma 4.2.** The following bounds hold: for all *i* = 1, . . . , *N*, for all *t* ≥ *t*₀,
+
+$$
+|\alpha'_{\pm,i}(t) \mp e_i \alpha_{\pm,i}(t)| \le C \left( e^{-4\sigma t} \|Z\|_{H^1\times L^2} + \|Z\|_{H^1\times L^2}^2 \right).
+$$

samples/texts/7376768/page_21.md

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+*Proof.* The proof is in a similar fashion as that of Claim 3.7. We note that
+
+$$
+\partial_t Z = \begin{pmatrix} 0 & Id \\ \partial_x^2 - Id + f'(\varphi) & 0 \end{pmatrix} Z + \begin{pmatrix} 0 \\ f(u) - f(\varphi) - (u-\varphi)f'(\varphi) \end{pmatrix}
+$$
+
+and that $|f(u) - f(\varphi) - (u-\varphi)f'(\varphi)| \le C|u-\varphi|^2 \le C\|Z\|_{H^{1\times L^2}}^2$. Thus for $i=1, \dots, N$, we have
+
+$$
+\begin{align*}
+a'_{\pm,i} &= \langle \partial_t Z, Z_{\pm,i} \rangle + \langle Z, \partial_t Z_{\pm,i} \rangle \\
+&= \left( \left( \begin{pmatrix} 0 & Id \\ \partial_x^2 - Id + f'(\varphi) & 0 \end{pmatrix} Z, Z_{\pm,i} \right) - \beta_i \langle Z, \partial_x Z_{\pm,i} \rangle + O(\|Z\|_{H^{1\times L^2}}^2) \right) \\
+&= \left( Z, \begin{pmatrix} -\beta_i \partial_x & \partial_x^2 - Id + f'(Q_i) \\ Id & -\beta_i \partial_x \end{pmatrix} Z_{\pm,i} \right) + \left( Z, \begin{pmatrix} 0 & f'(\varphi) - f'(Q_i) \\ 0 & 0 \end{pmatrix} Z_{\pm,i} \right) \\
+&\quad + O(\|Z\|_{H^{1\times L^2}}^2) \\
+&= \langle Z, \pm e_i Z_{\pm,i} \rangle + O(e^{-4\sigma t} \|Z\|_{H^{1\times L^2}} + \|Z\|_{H^{1\times L^2}}^2).
+\end{align*}
+$$
+
+**4.1.3. Control of the remaining modulation parameters.**
+
+**Lemma 4.3.** For all *i* = 1, ..., *N*, we have
+
+$$
+(4.6) \qquad |a'_i| \le C (\|E\|_{H^{1\times L^2}} + \|Z\|_{H^{1\times L^2}}^2).
+$$
+
+*Proof.* We do not detail the proof of this lemma which is similar to Claim 3.12. It suffices to start by differentiating the orthogonality relation $\langle E, (R_i)_x \rangle = 0$ with respect to $t$ and then to control terms by means of $\|E\|_{H^{1\times L^2}}$.
+
+**4.1.4. Study of a Lyapunov functional.** Taking some inspiration in [20, 29, 30], we consider for all $t \ge t_0$:
+
+$$
+F_z(t) := \int_{\mathbb{R}} \{\partial_x z^2 + \partial_t z^2 + z^2 - f'(\varphi)z^2\} dx + 2 \int_{\mathbb{R}} \chi \partial_x z \partial_t z dx,
+$$
+
+where $\chi$ is defined as follows.
+
+To begin with, recall that the parameters are ordered in such a way: $-1 < \beta_{\eta(1)} < \cdots < \beta_{\eta(N)} < 1$; let us denote, for some small $\delta > 0$ which will be determined later:
+
+$$
+\bar{l}_i := \beta_{\eta(i)} + \delta (\beta_{\eta(i+1)} - \beta_{\eta(i)})
+$$
+
+$$
+\underline{l}_i := \beta_{\eta(i)} - \delta (\beta_{\eta(i+1)} - \beta_{\eta(i)}).
+$$
+
+We then define for all $t \ge t_0$ and for all $x \in \mathbb{R}$:
+
+$$
+\chi(t,x) := \begin{cases}
+\beta_{\eta(1)} & \text{if } x \in (-\infty, \bar{l}_1 t] \\
+\beta_{\eta(i)} & \text{if } x \in [\underline{l}_i t, \bar{l}_i t] \\
+\beta_{\eta(N)} & \text{if } x \in [\bar{l}_N t, +\infty) \\
+\dfrac{x}{(1-2\delta)t} - \dfrac{\delta}{1-2\delta}(\beta_{\eta(i)} + \beta_{\eta(i+1)}) & \text{if } x \in [\bar{l}_i t, \underline{l}_{i+1} t], i \in \{1, \dots, N-1\}.
+\end{cases}
+$$
+
+For all $t \ge t_0$, $\chi(t)$ is a piecewise $\mathscr{C}^1$ function.
+
+Set $\Omega(t) := \bigcup_{i=1}^{N} (\bar{l}_i t, \underline{l}_{i+1} t)$. It follows from the definition of $\chi$ that
+
+$$
+\begin{gather*}
+\partial_t \chi(t,x) = \partial_x \chi(t,x) = 0 && \text{if } x \in \Omega(t)^\mathcal{C} \\
+\partial_x \chi(t,x) = \frac{1}{(1-2\delta)t}, && \partial_t \chi(t,x) = -\frac{x}{(1-2\delta)t^2} && \text{if } x \in \Omega(t).
+\end{gather*}
+$$

samples/texts/7376768/page_22.md

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+**Lemma 4.4.** There exists $\gamma > 0$ such that
+
+$$
+\begin{equation}
+\begin{aligned}
+\mathcal{F}_z'(t) ={}& 2 \int_{\Omega(t)} \partial_x z \partial_t z \partial_t \chi \, dx - \int_{\Omega(t)} \left\{ (\partial_t z)^2 + (\partial_x z)^2 - z^2 + f'(\varphi) z^2 \right\} \partial_x \chi \, dx \\
+& + O \left( e^{-\gamma t} \|Z\|_{H^{1} \times L^2}^2 + \|Z\|_{H^{1} \times L^2}^3 \right).
+\end{aligned}
+\end{equation}
+$$
+
+*Proof.* We essentially have to use the identity $\partial_t^2 z = \partial_x^2 z - z + f(u) - f(\varphi)$ in the expression of $\mathcal{F}_z'(t)$. We compute
+
+$$
+\begin{align}
+\mathcal{F}_z'(t) &= 2 \int_{\mathbb{R}} \left\{ z_t x z_x + z_{tt} z_t + z_t z - f'(\varphi) z_t z - \frac{1}{2} \varphi_t f''(\varphi) z^2 \right\} dx \nonumber \\
+&\quad + 2 \int_{\mathbb{R}} \chi_t z_x z_t \, dx + 2 \int_{\mathbb{R}} \chi z_x z_{tt} \, dx + 2 \int_{\mathbb{R}} \chi z_{xt} z_t \, dx \tag{4.8} \\
+&= 2 \int_{\mathbb{R}} z_t (-z_{xx} + z_{tt} + z - f'(\varphi)z) \, dx + 2 \int_{\mathbb{R}} \chi_t z_x z_t \, dx \nonumber \\
+&\quad + 2 \int_{\mathbb{R}} \chi z_x (z_{xx} - z + f(u) - f(\varphi)) \, dx - \int_{\mathbb{R}} z_t^2 \chi_x \, dx - \int_{\mathbb{R}} \varphi_t f''(\varphi) z^2 \, dx \nonumber
+\end{align}
+$$
+
+Notice that
+
+$$
+(4.9) \quad \int_{\mathbb{R}} z_t(-z_{xx} + z_{tt} + z - f'(\varphi)z) \, dx = \int_{\mathbb{R}} z_t(f(u) - f(\varphi) - f'(\varphi)z) \, dx = O(\|Z\|_{H^1 \times L^2}^3)
+$$
+
+and
+
+$$
+(4.10) \quad \int_{\mathbb{R}} \chi z_x (f(u) - f(\varphi)) \, dx = \int_{\mathbb{R}} \chi z_x f'(\varphi) z \, dx + O(\|Z\|_{H^1 \times L^2}^3).
+$$
+
+Hence, collecting (4.8), (4.9), and (4.10),
+
+$$
+\begin{align}
+\mathcal{F}_z'(t) &= 2 \int_{\mathbb{R}} \chi_t z_x z_t \, dx - \int_{\mathbb{R}} \chi_x (z_x^2 + z_t^2 - z^2) \, dx \notag \\
+&\quad - \int_{\mathbb{R}} z^2 (\chi_x f'(\varphi) + \chi_\varphi x f''(\varphi)) \, dx - \int_{\mathbb{R}} \varphi_t f''(\varphi) z^2 \, dx + O(\|Z\|_{H^{1}\times L^2}^3) \tag{4.11} \\
+&= 2 \int_{\mathbb{R}} z_x z_t \chi_t \, dx - \int_{\mathbb{R}} \{z_x^2 + z_t^2 - z^2 + f'(\varphi)z^2\} \chi_x \, dx \notag \\
+&\quad - \int_{\mathbb{R}} z^2 f''(\varphi)(\varphi_t + \chi_\varphi x) \, dx + O(\|Z\|_{H^{1}\times L^2}^3). \notag
+\end{align}
+$$
+
+Lastly, observe that
+
+$$
+\begin{align*}
+& \int_{\mathbb{R}} z^2 f''(\varphi)(\varphi_t + \chi_\varphi x) \, dx = \sum_{i=1}^{N} \int_{\mathbb{R}} z^2 f''(\varphi)((R_k)_t + \chi(R_k)_x) \, dx + O(\|Z\|_{H^{1}\times L^2}^3) \\
+& = I + J + O(\|Z\|_{H^{1}\times L^2}^3),
+\end{align*}
+$$
+
+where
+
+$$
+\begin{cases}
+I = \sum_{k=1}^{N} \int_{\Omega(t)} z^2 f''(\varphi) ((R_k)_t + \chi(R_k)_x) dx \\
+J = \sum_{k=1}^{N} \sum_{i=1}^{N} \int_{L_i(t)} l_i^2 z^2 f''(\varphi) ((R_k)_t + \chi(R_k)_x) dx.
+\end{cases}
+$$

samples/texts/7376768/page_23.md

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+On the one hand,
+
+$$
+\begin{align*}
+J &= \sum_{i=1}^{N} \sum_{k=1}^{N} \int_{L_i(t)}^{\bar{l}_i t} z^2 f''(\varphi) ((R_k)_t + \chi(R_k)_x) \, dx \\
+&= \sum_{i=1}^{N} \int_{L_i(t)}^{\bar{l}_i t} z^2 f''(\varphi) ((R_{\eta(i)})_t + \chi(R_{\eta(i)})_x) \, dx + O\left(e^{-4\sigma t} \|Z\|_{H^1 \times L^2}^2\right) \\
+&= O\left(e^{-4\sigma t} \|Z\|_{H^1 \times L^2}^2\right).
+\end{align*}
+$$
+
+Indeed, for all $x \in [\bar{l}_i(t), \bar{l}_i t]$, we have $(R_{\eta(i)})_t(t,x) + \chi(t,x)(R_{\eta(i)})_x(t,x) = 0$.
+
+On the other, for $x \in \Omega(t)$, there exists $k' \in \{1, \dots, N\}$ such that $\bar{l}_{k'}t \le x \le \underline{l}_{k'+1}t$.
+
+Then
+
+$$
+(\bar{l}_{k'} - \beta_k)t \leq x - \beta_k t \leq (\underline{l}_{k'+1} - \beta_k)t
+$$
+
+and thus
+
+$$
+|x - \beta_k t| \geq \delta \min_{j=1,\dots,N-1} \{\beta_{\eta(j+1)} - \beta_{\eta(j)}\} t > 0.
+$$
+
+As a consequence, we have for all $x \in \Omega(t)$
+
+$$
+\begin{equation}
+\begin{aligned}
+|(R_k)_x(t,x)| &\le Ce^{-\sigma|x-\beta_k t|} \\
+&\le Ce^{-\gamma t},
+\end{aligned}
+\tag{4.13}
+\end{equation}
+$$
+
+where $0 < \gamma < \sigma\delta \min_{j=1,...,N-1}\{\beta_{\eta(j+1)} - \beta_{\eta(j)}\}$.
+
+Hence
+
+$$
+(4.14) \qquad |I| \le C e^{-\gamma t} \|Z\|_{H^1 \times L^2}^2.
+$$
+
+Finally we gather (4.11), (4.12), and (4.14) in order to obtain (4.7). $\square$
+
+Let us introduce the components $\epsilon$ and $\epsilon_2$ of the vector
+
+$$
+E = \begin{pmatrix} \epsilon := z - \sum_i \{a_i \partial_x Q_i + b_i (Y_{+,i})_1\} \\ \epsilon_2 := \partial_t z - \sum_i \{a_i (-\beta_t \partial_{xx} Q_i) + b_t (Y_{+,i})_2\} \end{pmatrix}.
+$$
+
+**Corollary 4.5.** We have
+
+$$
+(4.15) \quad \mathcal{F}_z'(t) = 2 \int_{\Omega(t)} \epsilon_x \epsilon_2 \chi_t \, dx - \int_{\Omega(t)} \{ \epsilon_2^2 + \epsilon_x^2 - \epsilon^2 + f'(\varphi) \epsilon^2 \} \chi_x \, dx \\
+\phantom{(4.15) \quad \mathcal{F}_z'(t) = 2 \int_{\Omega(t)} } + O\left( e^{-\gamma t} \|Z\|_{H^1 \times L^2}^2 + \|Z\|_{H^1 \times L^2}^3 \right).
+$$
+
+*Proof.* The corollary immediately follows from (4.7) and bounds for the derivatives of $R_k$ and $Y_{+,k}$ which are analogous to (4.13). $\square$
+
+In the spirit of [20, Proposition 4.2], we will state an almost monotonicity property satisfied by $\mathcal{F}_z$. Let us define
+
+$$
+\mathcal{F}_{\varepsilon, \Omega(t)}(t) := \int_{\Omega(t)} \{\varepsilon_x^2 + \varepsilon_2^2 + 2\chi\varepsilon_x\varepsilon_2\} dx.
+$$
+
+Let $\lambda \in (1, \alpha - 1)$. The choice of $\lambda$ is linked with the integrability of particular quantities and will appear naturally later.
+
+**Proposition 4.6.** There exists $\delta > 0$ and $t_0 > 0$ such that for all $t \ge t_0$,
+
+$$
+(4.16) \qquad -\mathcal{F}'_z(t) \leq \frac{\lambda}{t} \mathcal{F}_{\varepsilon, \Omega(t)}(t) + O\left(e^{-\gamma t} \|Z\|_{H^1 \times L^2}^2 + \|Z\|_{H^1 \times L^2}^3\right).
+$$

samples/texts/7376768/page_24.md

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+*Proof.* Since $f'(0) = 0$ and $f'$ is continuous, there exists $r_0 > 0$ such that for all $r \in [0, r_0]$, $|f'(r)| \le 1$. There exists $K > 0$ (independent of $t$) such that for all $t \ge t_0$ and for all $x \in \Omega(t)$, $\lvert\varphi(t,x)\rvert \le Ke^{-\gamma t}$. Even if it means increasing $t_0$, we can assume that $Ke^{-\gamma t} \le r_0$ for all $t \ge t_0$.
+
+In addition $\chi_x \ge 0$. Thus, for $t \ge t_0$,
+
+$$ 
+\begin{aligned}
+& -2 \int_{\Omega(t)} \varepsilon_x \varepsilon_2 \chi_t \, dx + \int_{\Omega(t)} \{\varepsilon_x^2 + \varepsilon_2^2 - \varepsilon^2 + f'(\varphi)\varepsilon^2\} \chi_x \, dx \\
+& \le -2 \int_{\Omega(t)} \varepsilon_x \varepsilon_2 \chi_t \, dx + \int_{\Omega(t)} \{\varepsilon_x^2 + \varepsilon_2^2\} \chi_x \, dx.
+\end{aligned}
+$$
+
+Moreover,
+
+$$ 
+\begin{aligned}
+-2 \int_{\Omega(t)} \varepsilon_x \varepsilon_2 \chi_t \, dx + \int_{\Omega(t)} \{\varepsilon_x^2 + \varepsilon_2^2\} \chi_x \, dx &= \frac{1}{(1-2\delta)t} \int_{\Omega(t)} \{2\varepsilon_x\varepsilon_2 \frac{x}{t} + \varepsilon_x^2 + \varepsilon_2^2\} \, dx \\
+&= \frac{1}{(1-2\delta)t} \left( \mathcal{F}_{\varepsilon, \Omega(t)} + 2 \int_{\Omega(t)} \left(\frac{x}{t} - \chi\right) \varepsilon_x \varepsilon_2 \, dx \right).
+\end{aligned}
+$$
+
+Now, for $x \in \Omega(t)$, we have
+
+$$ 
+\begin{aligned}
+\left|\frac{x}{t} - \chi(t,x)\right| &\leq \left|\frac{x}{t}\left(1-\frac{1}{1-2\delta}\right)\right| + \frac{\delta}{1-2\delta} \times 2 \max_i |\beta_i| \\
+&\leq 2\delta \left(\left|\frac{x}{t}\right| + C\right) \\
+&\leq C\delta.
+\end{aligned}
+$$
+
+Thus,
+
+$$ 
+\frac{2}{(1-2\delta)t} \int_{\Omega(t)} \left(\frac{x}{t} - \chi\right) \varepsilon_x \varepsilon_2 \, dx = O\left(\delta \|E\|_{H^{1}\times L^2}\right).
+$$
+
+Noticing moreover that
+
+$$ 
+\begin{aligned}
+\mathcal{F}_{\varepsilon, \Omega(t)} &\geq \int_{\Omega(t)} (\varepsilon_x^2 + \varepsilon_2^2) \, dx - 2\|\chi(t)\|_{\infty} \int_{\Omega(t)} \varepsilon_x \varepsilon_2 \, dx \\
+&\geq \int_{\Omega(t)} (\varepsilon_x^2 + \varepsilon_2^2) \, dx - \|(\chi(t))'\|_{\infty} \int_{\Omega(t)} (\varepsilon_x^2 + \varepsilon_2^2) \, dx \\
+&\geq (1 - \|(\chi(t))'\|_{\infty}) \|E\|_{H^{1}\times L^2(\Omega(t))}^2,
+\end{aligned}
+$$
+
+we obtain
+
+$$ 
+\frac{2}{(1-2\delta)t} \int_{\Omega(t)} \left(\frac{x}{t} - \chi\right) \epsilon_x \epsilon_2 dx = O(\delta F_{\epsilon, \Omega(t)}).
+$$
+
+Finally,
+
+$$ 
+\begin{aligned}
+-F'_z(t) &\leq \frac{1}{(1-2\delta)t}(1+\mathcal{O}(\delta))F_{\epsilon,\Omega(t)} + O(e^{-\gamma t} \|Z\|_{H^{1}\times L^2}^2 + \|Z\|_{H^{1}\times L^2}^3) \\
+&\leq \frac{\lambda}{t} F_{\epsilon,\Omega(t)} + O(e^{-\gamma t} \|Z\|_{H^{1}\times L^2}^2 + \|Z\|_{H^{1}\times L^2}^3),
+\end{aligned}
+$$
+
+provided $\delta$ is chosen small enough. $\square$
+
+We now introduce the functional
+
+$$ 
+F_\epsilon(t) := \int_{\mathbb{R}} (\epsilon_x^2 + \epsilon_2^2 + \epsilon^2 - f'(\varphi)\epsilon^2) dx + 2 \int_{\mathbb{R}} \chi \epsilon_x \epsilon_2 dx
+$$
+
+and compare it with $F_z$.
+
+**Lemma 4.7.** We have
+
+$$ 
+F_\epsilon(t) = F_z(t) - 2 \sum_{i=1}^{N} \alpha_{-,i}(t) \alpha_{+,i}(t) + G(t),
+$$

samples/texts/7376768/page_25.md

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+where $\mathcal{G}(t) = O(e^{-\gamma t} \|Z\|_{H^1 \times L^2}^2)$ and $\mathcal{G}'(t) = O(e^{-\gamma t} \|Z\|_{H^1 \times L^2}^2)$.
+
+*Proof.* Let $M$ be the matrix $\begin{pmatrix} -\partial_x^2 + Id - f'(\varphi) & 0 \\ 0 & Id \end{pmatrix}$. For all $i=1, \dots, N$, we have the decomposition
+
+$$ (4.17) \qquad M = H_i + \begin{pmatrix} 0 & \beta_i \partial_x \\ -\beta_i \partial_x & 0 \end{pmatrix} + \begin{pmatrix} f'(Q_i) - f'(\varphi) & 0 \\ 0 & 0 \end{pmatrix}. $$
+
+Then we infer
+
+$$ 
+\begin{align*}
+\mathcal{F}_\varepsilon(t) &= \langle ME, E \rangle + 2 \int_{\mathbb{R}} \chi \varepsilon_x \varepsilon_2 \, dx \\
+&= \langle MZ, Z \rangle - 2 \left\langle M \left( \sum_{i=1}^N \{a_i(R_i)_x + b_i Y_{+,i}\} \right), Z \right\rangle \\
+&\quad + \left\langle M \left( \sum_{i=1}^N \{a_i(R_i)_x + b_i Y_{+,i}\} \right), \sum_{j=1}^N \{a_j(R_j)_x + b_j Y_{+,j}\} \right\rangle \\
+&\quad + 2 \int_{\mathbb{R}} \chi z_x z_t \, dx + 2 \int_{\mathbb{R}} \chi \sum_{i=1}^N \{a_i(Q_i)_{xx} + b_i(Y_{+,i})_{1,x}\} \sum_{j=1}^N \{a_j(Q_j)_{xt} + b_j(Y_{+,j})_2\} \, dx \\
+&\quad - 2 \int_{\mathbb{R}} \chi z_x \sum_{i=1}^N \{a_i(Q_i)_{xt} + b_i(Y_{+,i})_2\} \, dx - 2 \int_{\mathbb{R}} \chi z_t \sum_{i=1}^N \{a_i(Q_i)_{xx} + b_i(Y_{+,i})_{1,x}\} \, dx \\
+&= \mathcal{F}_z(t) - 2 \sum_{i=1}^N b_i \langle Z_{+,i}, Z \rangle + \tilde{\mathcal{G}}(t) \\
+&= \mathcal{F}_z(t) - 2 \sum_{i=1}^N \alpha_{-,i}(t) \alpha_{+,i}(t) + \mathcal{G}(t),
+\end{align*}
+$$
+
+with
+
+$$ 
+\begin{align*}
+MY_{+,i} &= Z_{+,i} + \beta_i \begin{pmatrix} (Y_{+,i})_{2,x} \\ -(Y_{+,i})_{1,x} \end{pmatrix} + \begin{pmatrix} (f'(Q_i) - f'(\varphi))(Y_{+,i})_1 \\ 0 \end{pmatrix} \\
+M(R_i)_x &= \begin{pmatrix} -\beta_i^2 (Q_i)_{xxx} \\ -\beta_i (Q_i)_{xx} \end{pmatrix} + \begin{pmatrix} (f'(Q_i) - f'(\varphi))(Q_i)_x \\ 0 \end{pmatrix}.
+\end{align*}
+$$
+
+and
+
+$$ 
+\begin{align*}
+\tilde{\mathcal{G}}(t) ={}& -2 \left\langle \sum_{i=1}^{N} \left\{ a_i \begin{pmatrix} -\beta_i^2 (Q_i)_{xxx} \\ \beta_i (Q_i)_{xx} \end{pmatrix} + b_i \beta_i \begin{pmatrix} (Y_{+,i})_{2,x} \\ -(Y_{+,i})_{1,x} \end{pmatrix} \right\}, Z \right\rangle + O(e^{-\gamma t} \|Z\|_{H^1 \times L^2}^2) \\
+&+ \sum_{i=1}^{N} \left\langle a_i \begin{pmatrix} -\beta_i^2 (Q_i)_{xxx} \\ \beta_i (Q_i)_{xx} \end{pmatrix} + b_i \beta_i \begin{pmatrix} (Y_{+,i})_{2,x} \\ -(Y_{+,i})_{1,x} \end{pmatrix} + b_i Z_{+,i}, a_i (R_i)_x + b_i Y_{+,i} \right\rangle \\
+&+ 2 \sum_{i=1}^{N} \int_{\mathbb{R}} \chi (a_i (Q_i)_{xx} + b_i (Y_{+,i})_{1,x}) (-\beta_i a_i (Q_i)_{xx} + b_i (Y_{+,i})_2) \, dx \\
+&- 2 \sum_{i=1}^{N} \int_{\mathbb{R}} \chi z_x (-\beta_i a_i (Q_i)_{xx} + b_i (Y_{+,i})_2) \, dx - 2 \sum_{i=1}^{N} \int_{\mathbb{R}} \chi z_t (a_i (Q_i)_{xx} + b_i (Y_{+,i})_{1,x}) \, dx.
+\end{align*}
+$$

samples/texts/7376768/page_26.md

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+Integrating by parts, we obtain
+
+$$
+\begin{align*}
+& \int_{\mathbb{R}} \chi z_x (-\beta_i a_i (Q_i)_{xx} + b_i (Y_{+,i})_2) \, dx \\
+&= - \int_{\mathbb{R}} \chi z (-\beta_i a_i (Q_i)_{xxx} + b_i (Y_{+,i})_{2,x}) \, dx - \int_{\mathbb{R}} \chi_x z (-\beta_i a_i (Q_i)_{xx} + b_i (Y_{+,i})_2) \, dx.
+\end{align*}
+$$
+
+Consequently,
+
+$$
+\begin{align*}
+\tilde{\mathcal{G}}(t) &= -2 \sum_{i=1}^{N} a_i \int_{\mathbb{R}} (\chi - \beta_i) (-\beta_i z(Q_i)_{xxx} + z_t(Q_i)_{xx}) \, dx \\
+&\quad - 2 \sum_{i=1}^{N} b_i \int_{\mathbb{R}} (\chi - \beta_i) (-z(Y_{+,i})_{2,x} + z_t(Y_{+,i})_{1,x}) \, dx \\
+&\quad + 2 \sum_{i=1}^{N} a_i^2 \beta_i \int_{\mathbb{R}} (\beta_i - \chi)(Q_i)_{xx}^2 \, dx + 2 \sum_{i=1}^{N} b_i^2 \int_{\mathbb{R}} (\beta_i - \chi)(Y_{+,i})_1 (Y_{+,i})_{2,x} \\
+&\quad + 2 \sum_{i=1}^{N} \int_{\Omega(t)} \chi_x z (-\beta_i a_i (Q_i)_{xx} + b_i (Y_{+,i})_2) \, dx \\
+&\quad + 2 \sum_{i=1}^{N} a_i b_i \int_{\mathbb{R}} (\chi - \beta_i) (-\beta_i (Y_{+,i})_{1,x}(Q_i)_{xx} + (Q_i)_{xx}(Y_{+,i})_2) \, dx + O(e^{-\gamma t} \|Z\|_{H^{1\times L^2}}^2) \\
+&= O(e^{-\gamma t} \|Z\|_{H^{1\times L^2}}^2).
+\end{align*}
+$$
+
+□
+
+We deduce from Proposition 4.6 and Lemma 4.7 the following "weak" monotonicity property.
+
+**Corollary 4.8.** We have for all $t \geq t_0$,
+
+$$
+(4.18) \quad - \mathcal{F}_{\varepsilon}'(t) \leq \frac{\lambda}{t} \mathcal{F}_{\varepsilon}(t) + \frac{C}{t} \sum_{i=1}^{N} \alpha_{+,i}^2 + O \left( e^{-\gamma t} \|Z\|_{H^{1\times L^2}}^2 + \|Z\|_{H^{1\times L^2}}^3 \right).
+$$
+
+Proof. From Lemma 4.2, we obtain: for all $i = 1, \dots, N$, for all $t \ge t_0$,
+
+$$
+(4.19) \quad |(\alpha_{-,i} \alpha_{+,i})'| \le C (e^{-\gamma t} \|Z\|_{H^{1\times L^2}}^2 + \|Z\|_{H^{1\times L^2}}^3).
+$$
+
+Thus, we have
+
+$$
+- \mathcal{F}_{\epsilon}'(t) = - \mathcal{F}_z'(t) + 2 \sum_{i=1}^{N} (\alpha_{-,i} \alpha_{+,i})' + O(e^{-\gamma t} \|Z\|^2) \\
+\leq \frac{\lambda}{t} \mathcal{F}_{\epsilon, \Omega(t)} + O(e^{-\gamma t} \|Z\|_{H^{1\times L^2}}^2 + \|Z\|_{H^{1\times L^2}}^3).
+$$
+
+We now make use of the following property satisfied by $\mathcal{F}_\epsilon$ which is a consequence of a localized version of Proposition 2.1 (we refer to [20, proof of (4.12) and (4.21)] for similar considerations in the case of the energy-critical wave equation):
+
+$$
+(4.20) \qquad \mathcal{F}_{\epsilon, \Omega(t)} \leq \mathcal{F}_{\epsilon}(t) + C \sum_{i=1}^{N} \langle \epsilon, Z_{+,i} \rangle^2,
+$$
+
+to deduce that
+
+$$
+- \mathcal{F}_{\varepsilon}'(t) \leq \frac{\lambda}{t} \mathcal{F}_{\varepsilon}(t) + \frac{C}{t} \sum_{i=1}^{N} a_{+,i}^2 + O\left(e^{-\gamma t} \|Z\|_{H^{1\times L^2}}^2 + \|Z\|_{H^{1\times L^2}}^3\right). \quad \square
+$$

samples/texts/7376768/page_27.md

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+We are now in a position to prove
+
+**Proposition 4.9.** Even if it means taking a larger $t_0$, we have for all $t \ge t_0$
+
+$$
+(4.21) \qquad \|Z(t)\|_{H^1 \times L^2} \le C \sup_{t' \ge t} \sum_{i=1}^{N} |\alpha_{-,i}(t)|.
+$$
+
+Proof. Multiplying the estimate obtained in Corollary 4.8 by $t^\lambda$, we have for all $t \ge t_0$,
+
+$$
+-(t^\lambda \mathcal{F}_E)'(t) \le C \left( \sum_{i=1}^{N} t^{\lambda-1} \alpha_{+,i}^2 + t^\lambda e^{-\gamma t} \|Z\|_{H^{1}\times L^2}^2 + t^\lambda \|Z\|_{H^{1}\times L^2}^3 \right).
+$$
+
+Since, $t^{\lambda-1}\alpha_{+,i}^2, t^\lambda \|Z\|_{H^1 \times L^2}^3$ are integrable functions of $t$ on $[t_0, +\infty)$ and $t^\lambda \mathcal{F}_E(t) \to 0$ as $t \to +\infty$, we infer that
+
+$$
+(4.22) \qquad \mathcal{F}_E(t) \le C \left( \frac{1}{t^\lambda} \sum_{i=1}^N \int_t^{+\infty} t^{\lambda-1} \alpha_{+,i}^2(t') dt' + \frac{C}{t^\lambda} \int_t^{+\infty} t^{\lambda} \|Z(t')\|_{H^1 \times L^2}^3 dt' + e^{-\gamma t} \sup_{t' \ge t} \|Z(t')\|^2 \right).
+$$
+
+By the coercivity property satisfied by $\mathcal{F}_E$, we thus obtain
+
+$$
+\begin{align*}
+\|E(t)\|_{H^1 \times L^2}^2 &\le C \left( \mathcal{F}_E(t) + \sum_{i=1}^N \alpha_{+,i}^2(t) + e^{-2\gamma t} \|Z(t)\|_{H^1 \times L^2}^2 \right) \\
+&\le \frac{C}{t^\lambda} \sum_{i=1}^N \int_t^{+\infty} t^{\lambda-1} \alpha_{+,i}^2(t') dt' + \frac{C}{t^\lambda} \int_t^{+\infty} t^{\lambda'} \|Z(t')\|_{H^1 \times L^2}^3 dt' \\
+&\quad + C e^{-\gamma t} \sup_{t' \ge t} \|Z(t')\|_{H^1 \times L^2}^2 + C \sum_{i=1}^N \alpha_{+,i}^2(t).
+\end{align*}
+$$
+
+In other words, there exists $C \ge 0$ such that for all $t \ge t_0$,
+
+$$
+(4.23) \qquad \|E(t)\|_{H^1 \times L^2} \le \frac{C}{t^{\frac{1}{2}}} \left( \int_t^{+\infty} t^{\lambda-1} \sum_{i=1}^{N} \alpha_{+,i}^2(t') dt' \right)^{\frac{1}{2}} + C \left( \sum_{i=1}^{N} \alpha_{+,i}^2(t) \right)^{\frac{1}{2}} \\
+\qquad\qquad + \frac{C}{t^{\frac{1}{2}}} \left( \int_t^{+\infty} t^{\lambda'} \|Z(t')\|_{H^{1}\times L^2}^3 dt' \right)^{\frac{1}{2}} + C e^{-\frac{\gamma}{2}t} \sup_{t'\ge t} \|Z(t')\|_{H^{1}\times L^2}.
+$$
+
+From Lemma 4.2 we recall the estimate
+
+$$
+|\alpha'_{+,i}(t) - e_i \alpha_{+,i}(t)| \le C (e^{-\gamma t} \|Z(t)\|_{H^1 \times L^2} + \|Z(t)\|_{H^1 \times L^2}^2)
+$$
+
+which is equivalent to
+
+$$
+|(e^{-e_i t} \alpha_{+,i})'(t)| \le C e^{-e_i t} (e^{-\gamma t} \|Z(t)\|_{H^1 \times L^2} + \|Z(t)\|_{H^1 \times L^2}^2).
+$$
+
+Integrating the preceding inequality (which is indeed possible), we deduce:
+
+$$
+(4.24) \qquad |α_{+,i}(t)| \le C \left( e^{-γt} \sup_{t'≥t} ||Z(t')||_{H^1×L^2} + \sup_{t'≥t} ||Z(t')||_{H^1×L^2}^2 \right).
+$$
+
+Now,
+
+$$
+\|Z(t)\|_{H^1 \times L^2} \le \|E(t)\|_{H^1 \times L^2} + C \sum_{i,\pm} |\alpha_{\pm,i}(t)| + C \sum_{i=1}^{N} |a_i(t)| + C e^{-\gamma t} \|Z(t)\|_{H^1 \times L^2}.
+$$
+
+By Lemma 4.3, it follows that
+
+$$
+(4.25) \qquad \|Z(t)\|_{H^1 \times L^2} \le C \left( \|E(t)\|_{H^1 \times L^2} + \sum_{\pm, i} |\alpha_{\pm, i}(t)| + \int_t^{+\infty} (\|E(t')\|_{H^1 \times L^2} + \|Z(t')\|_{H^1 \times L^2}) dt' \right).
+$$

samples/texts/7376768/page_28.md

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+Observing that the quantity $\int_t^{+\infty} \|Z(t')\|_{H^1 \times L^2} dt'$ makes sense and tends to 0 as $t \to +\infty$ (because $\alpha > 1$) and that
+
+$$ \int_t^{+\infty} \|Z(t')\|_{H^1 \times L^2}^2 dt' \leq \sup_{t' \geq t} \|Z(t')\|_{H^1 \times L^2} \int_t^{+\infty} \|Z(t')\|_{H^1 \times L^2} dt, $$
+
+we deduce that for $t$ sufficiently large
+
+$$ (4.26) \quad \|Z(t)\|_{H^1 \times L^2} \le C \left( \sup_{t' \ge t} \|E(t')\|_{H^1 \times L^2} + \int_t^{+\infty} \|E(t')\|_{H^1 \times L^2} dt' + \sup_{t' \ge t} \sum_{\pm, i} |\alpha_{\pm, i}(t')| \right). $$
+
+Now, we replace (4.24) in (4.23) and use (4.26). We notice that the following well-defined quantities tend to 0 as $t \to +\infty$ (because $\alpha > 3$ and by the choice of $\lambda < \alpha - 1$):
+
+$$ \begin{align*} & \int_t^{+\infty} t'^{\lambda-1} \sup_{t'' \ge t'} \|Z(t'')\|_{H^1 \times L^2}^2 dt', && \int_t^{+\infty} \sup_{t'' \ge t'} \|Z(t'')\|_{H^1 \times L^2}^2 dt' \\ & \int_t^{+\infty} t'^{\lambda} \|Z(t')\|_{H^1 \times L^2}^2 dt', && \int_t^{+\infty} \frac{1}{u^{\frac{4}{5}}} \left( \int_u^{+\infty} t'^{\lambda-1} \sup_{t'' \ge t'} \|Z(t'')\|_{H^1 \times L^2}^2 dt' \right)^{\frac{1}{2}} du \\ & \int_t^{+\infty} \frac{1}{u^{\frac{4}{5}}} \left( \int_u^{+\infty} t'^{\lambda} \|Z(t')\|_{H^1 \times L^2}^2 dt' \right)^{\frac{1}{2}} du. \end{align*} $$
+
+We then obtain for $t$ sufficiently large
+
+$$ \|Z(t)\|_{H^1 \times L^2} \le C \sup_{t' \ge t} \sum_{i=1}^{N} |\alpha_{-,i}(t')|. $$
+
+□
+
+**Proposition 4.10.** We have for all $t \ge t_0$, for all $i = 1, \dots, N$,
+
+$$ (4.27) \qquad |\alpha_{-,i}(t)| \le Ce^{-e_1 t} $$
+
+*Proof.* From Lemma 4.2 and Proposition 4.9, it results that for all $i = 1, \dots, N$, for all $t \ge t_0$,
+
+$$ (4.28) \qquad |\alpha'_{-,i}(t) + e_i \alpha_{-,i}(t)| \le C \left( e^{-\sigma t} \sup_{t' \ge t} \sum_{j=1}^{N} |\alpha_{-,j}(t')| + \left( \sup_{t' \ge t} \sum_{j=1}^{N} |\alpha_{-,j}(t')| \right)^2 \right). $$
+
+Then, for all $i = 1, \dots, N$,
+
+$$ (4.29) \qquad |\alpha'_{-,i}(t)\alpha_{-,i}(t) + e_i\alpha_{-,i}^2(t)| \le C \left( e^{-\sigma t} \sup_{t' \ge t} \sum_{j=1}^{N} |\alpha_{-,j}(t')||\alpha_{-,i}(t)| + \sup_{t' \ge t} \left( \sum_{j=1}^{N} |\alpha_{-,j}(t')| \right)^2 |\alpha_{-,i}(t)| \right). $$
+
+Let us denote $\mathcal{A} := \sum_{j=1}^N \alpha_{-,j}^2$. Summing on $i = 1, \dots, N$, we have in particular:
+
+$$ |\mathcal{A}'(t) + 2e_1\mathcal{A}(t)| \le C e^{-\sigma t} \left( \sup_{t' \ge t} \sum_{j=1}^{N} |\alpha_{-,j}(t')| \right) \sum_{i=1}^{N} |\alpha_{-,i}(t)| + C \left( \sup_{t' \ge t} \sum_{j=1}^{N} |\alpha_{-,j}(t')| \right)^2 \sum_{i=1}^{N} |\alpha_{-,i}(t)|. $$
+
+Noticing that $(\sum_{j=1}^N |\alpha_{-,j}|)^2 \le C\mathcal{A}$, we obtain the existence of $c > 0$ such that for all $t \ge t_0$,
+
+$$ |\mathcal{A}'(t) + 2e_1\mathcal{A}(t)| \le C(e^{-\sigma t} + (\mathcal{A}(t))^{1/2})\sup_{t' \ge t}\mathcal{A}(t'). $$
+
+Lastly we observe that $\xi: t \mapsto e^{-\sigma t} + \mathcal{A}(t)^{1/2}$ is integrable on $[t_0, +\infty)$ since
+
+$$ \mathcal{A}(t)^{\frac{1}{2}} = O\left(\|Z(t)\|_{H^1 \times L^2}^{\frac{1}{2}}\right) = O\left(\frac{1}{t^{\frac{\alpha}{2}}}\right). $$

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+By Lemma 7.1 in Appendix, we obtain $\mathcal{A}(t) \le Ce^{-2e_1 t}$ for $t \ge t_0$. Consequently, for all $i = 1, \dots, N$,
+$$|\alpha_{-,i}(t)| \le Ce^{-e_1 t}.$$
+
+Gathering Propositions 4.9 and 4.10, we deduce
+
+**Proposition 4.11.** There exists $C > 0$ such that for $t$ sufficiently large,
+
+$$\|Z(t)\|_{H^1 \times L^2} \le C e^{-e_1 t}.$$
+
+**4.2. Identification of the solution.** Recall that we have constructed in Section 3 a family of multi-solitons $(\phi_{A_1, \dots, A_N})$ such that for all $j=1, \dots, N$, for all $t \ge t_0$,
+
+$$ (4.30) \quad \| \Phi_{A_1, \dots, A_j}(t) - \Phi_{A_1, \dots, A_{j-1}}(t) - A_j e^{-e_j t} Y_{+,j}(t) \|_{H^1 \times L^2} \le C e^{-(e_j+\sigma)t}. $$
+
+(We can always assume that $\sigma < \min\{e_1, \min_{j=2,\dots,N}\{e_j - e_{j-1}\}\}$).
+
+Following the strategy of Combet [4], our goal is to establish
+
+**Proposition 4.12.** For all $j = 1, \dots, N$, there exist $C \ge 0$, $t_0 \ge 0$, and $A_1, \dots, A_j \in \mathbb{R}$ such that, defining
+
+$$E_j := U - \Phi_{A_1, \dots, A_j},$$
+
+we have:
+
+$$ (4.31) \qquad \|E_j(t)\|_{H^1 \times L^2} \le C e^{-e_j t}. $$
+
+Moreover, denoting $\alpha_{\pm,j,k} := \langle E_j, Z_{\pm,k} \rangle$ for all $k = 1, \dots, N$, we have
+
+$$ (4.32) \quad \forall k \in \{1, \dots, j\}, \quad e^{e_k t} \alpha_{-,j,k}(t) \to 0, \text{ as } t \to +\infty. $$
+
+*Proof.* We proceed by induction on $j$. First, we focus on the case where $j = 1$. We have $\|Z(t)\|_{H^1 \times L^2} \le C e^{-e_1 t}$ by Proposition 4.11. Thus, by Lemma 4.2 and given that $\sigma < e_1$,
+
+$$|(e^{e_1 t} \alpha_{-,1})'| \le C e^{-\sigma t}.$$
+
+Since $t \mapsto e^{-\sigma t}$ is integrable in $+\infty$, there exists $A_1 \in \mathbb{R}$ such that
+
+$$e^{e_1 t} \alpha_{-,1}(t) \to A_1, \quad \text{as } t \to +\infty.$$
+
+We then define $E_1 := U - \Phi_{A_1}$. We notice that $E_1 = E + (\Phi - \Phi_{A_1})$ so that
+
+$$
+\begin{align*}
+\|E_1(t)\|_{H^1 \times L^2} &\le \|E(t)\|_{H^1 \times L^2} + \|(\Phi - \Phi_{A_1})(t)\|_{H^1 \times L^2} \\
+&\le C e^{-e_1 t} + \|(\Phi_{A_1} - \Phi)(t) - A_1 e^{-e_1 t} Y_{+,1}(t)\|_{H^1 \times L^2} + \|A_1 e^{-e_1 t} Y_{+,1}(t)\|_{H^1 \times L^2} \\
+&\le C e^{-e_1 t}.
+\end{align*}
+$$
+
+Moreover
+
+$$
+\begin{align*}
+\alpha_{-,1,1} &= \langle E, Z_{-,1} \rangle + (\Phi - \Phi_{A_1} - A_1 e^{-e_1 t} Y_{+,1}, Z_{-,1}) + A_1 e^{-e_1 t} \langle Y_{+,1}, Z_{-,1} \rangle \\
+&= -\alpha_{-,1} + A_1 e^{-e_1 t} + O(e^{-(e_1+\sigma)t}) \\
+&= o(e^{-e_1 t}),
+\end{align*}
+$$
+
+the last line resulting from the definition of $A_1$.
+
+Thus Proposition 4.12 is true for $j = 1$.
+
+We now assume that there exist $A_1, \dots, A_{j-1} \in \mathbb{R}$ such that $\|E_{j-1}(t)\|_{H^1 \times L^2} \le C e^{-e_{j-1}t}$ and for all $k = 1, \dots, j-1$, $e^{e_k t} \alpha_{-,j-1,k}(t) \to 0$ as $t \to +\infty$.
+
+Let us show
+
+**Claim 4.13.** We have
+
+$$\|E_{j-1}(t)\|_{H^1 \times L^2} \le C e^{-e_j t}.$$

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+properties which are needed in this paper, as well as the introduction of useful notations, are presented
+in the next subsection. Note that a similar spectral theory was firstly considered by Pego and Weinstein
+[26] in the context of the generalized Korteweg-de Vries equations.
+
+Starting from this point of view, we are interested in solutions which converge to a soliton or a
+sum of solitons for large values of $t$; these solutions are classically known as *multi-solitons*.
+
+Let us consider an integer $N \ge 1$ and $2N$ parameters
+
+$x_1, \dots, x_N \in \mathbb{R}^d \text{ and } \beta_1, \dots, \beta_N \in \mathbb{R}^d$
+
+such that
+
+$$
+\forall i = 1, \dots, N, \quad |\beta_i| < 1 \quad \text{and} \quad \forall i \neq j, \quad \beta_i \neq \beta_j.
+$$
+
+We recall the following theorem by Côte and Muñoz which states the existence of at least one multi-soliton.
+
+**Theorem 1.1** ([9]). There exist $\sigma_0$, $t_0 \in \mathbb{R}$ and $C_0 > 0$, only depending on the sets $(\beta_i)_i$, $(x_i)_i$, and a solution $U = \begin{pmatrix} u \\ \partial_t u \end{pmatrix} \in \mathcal{C}([t_0, +\infty), H^1(\mathbb{R}^d) \times L^2(\mathbb{R}^d))$ of (NLKG) such that for all $t \ge t_0$,
+
+$$
+\left\| U(t) - \sum_{i=1}^{N} R_{\beta_i, x_i}(t) \right\|_{H^1 \times L^2} \le C_0 e^{-\sigma_0 t}.
+$$
+
+Let us mention that, dealing with complex valued solutions of (NLKG), thus opening the possibi-
+lity of considering stable solitons, Bellazzani, Ghimenti and Le Coz [1] obtained a similar existence
+result for (NLKG) in this particular stable case. We also notice that the previous theorem has been
+extended to solutions describing multi-bound states by Côte and Martel [7], that is to multi-traveling
+waves made of any number *N* of decoupled general (excited) bound states. In the present paper, we
+will however only focus on (real valued) multi-solitary waves in the above sense.
+
+Since solitons are unstable, a solution of (NLKG) which behaves as a soliton in large time is not
+expected to be necessarily a soliton. One of our goals is thus to precise the dynamic of the flow of
+(NLKG) near a soliton. Similarly, the dynamic near a sum of solitons is also supposed to be more
+complex as time goes to infinity.
+
+**1.2. Main results.** Given $N$ distinct velocity parameters, we aim at proving the existence of a whole family of multi-solitons which turns out to be the unique family of multi-solitons in a certain class of solutions. Our first result reads as follows.
+
+**Theorem 1.2.** Assume that $f$ is of class $\mathcal{C}^2$ and $0 < |\beta_N| < \dots < |\beta_1| < 1$. There exist $\sigma > 0$, $0 < e^{\beta_1} < \dots < e^{\beta_N}$, $Y_{+,i} \in \mathcal{C}(\mathbb{R}, H^1(\mathbb{R}^d) \times L^2(\mathbb{R}^d)) \cap L^\infty(\mathbb{R}, H^1(\mathbb{R}^d) \times L^2(\mathbb{R}^d))$ for $i = 1, \dots, N$ and an $N$-parameter family $(\varphi_{A_1, \dots, A_N})_{(A_1, \dots, A_N) \in \mathbb{R}^N}$ of solutions of (NLKG) such that, for all $(A_1, \dots, A_N) \in \mathbb{R}^N$, there exist $t_0 \in \mathbb{R}$ and $C > 0$ such that
+
+$$
+(1.2) \quad \forall t \ge t_0, \quad \left\| \Phi_{A_1, \ldots, A_N}(t) - \sum_{i=1}^{N} R_{\beta_i, x_i}(t) - \sum_{i=1}^{N} A_i e^{-e^{\beta_i} t} Y_{+,i}(t) \right\|_{H^{1} \times L^{2}} \le C e^{-(e^{\beta_N}+\sigma)t},
+$$
+
+where $\Phi_{A_1,...,A_N} := (\varphi_{A_1,...,A_N})_{\partial_t \varphi_{A_1,...,A_N}}$. In addition, if $(A'_1,...,A'_N) \neq (A_1,...,A_N)$, then $\varphi_{A'_1,...,A'_N} \neq \varphi_{A_1,...,A_N}$.
+
+*Remark 1.3.* The parameters $e_{\beta_i}$ and the functions $Y_{+,i}$ ($i = 1, \dots, N$) are defined in Proposition 2.1 and in subsection 2.2.
+
+One can moreover precise the value of $\sigma$ in Theorem 1.2; for this, we refer to (2.1).
+
+Our next result is concerned with the classification of multi-solitons. We aim at proving that any multi-soliton should belong to the family constructed in Theorem 1.2 above. This is indeed the case, if one knows that the multi-soliton converges sufficiently fast to its profile. As it is stated below, a decay in a power of *t* of degree larger than 3 is a sufficient rate.

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+* To prove this claim, we show that, if $\|E_{j-1}(t)\| \le C e^{-\sigma_0 t}$ with $e_{j-1} < \sigma_0 < e_j - \sigma$, then
+
+$$ \|E_{j-1}(t)\|_{H^1 \times L^2} \le C e^{-(\sigma+\sigma_0)t}. $$
+
+As
+
+$$ 
+\begin{aligned}
+|\alpha'_{\pm,j-1,k}(t) \mp e_k \alpha_{\pm,j-1,k}(t)| &\le C (e^{-\sigma t} \|E_{j-1}(t)\|_{H^1 \times L^2} + \|E_{j-1}(t)\|^2_{H^1 \times L^2}) \\
+&\le Ce^{-(\sigma+\sigma_0)t}
+\end{aligned}
+ $$
+
+(by the same calculations and arguments as those developed in the proof of Lemma 4.2), we
+have for all $k=1, \dots, j-1$,
+
+$$ |(e^{e_k t} \alpha_{-,j-1,k})'| \le C e^{-(\sigma+\sigma_0-e_k)t}. $$
+
+Since $t \mapsto e^{-(\sigma+\sigma_0-e_k)t}$ is integrable in the neighborhood of $+\infty$ (since $e_k \le e_{j-1}$), and by assumption, $e^{e_k t} \alpha_{-,j-1,k}(t) \to 0$ as $t \to +\infty$, we have by integration
+
+$$ |\alpha_{-,j-1,k}(t)| \le C e^{-(\sigma+\sigma_0)t}. $$
+
+For all $k = j, \dots, N$, we have $\sigma + \sigma_0 - e_k \le \sigma + \sigma_0 - e_j < 0$, thus by integration on $[t_0, t]$,
+we obtain
+
+$$ |e^{e_k t} \alpha_{-,j-1,k}(t) - e^{e_k t_0} \alpha_{-,j-1,k}(t_0)| \le C e^{(e_k - \sigma_0 - \sigma)t}. $$
+
+Eventually, we obtain (by a "cut-and-paste" of the argument exposed in subsection 4.1)
+
+$$ \|E_{j-1}(t)\|_{H^1 \times L^2} \le C \sup_{t' \ge t} \sum_{k=1}^{N} |\alpha_{-,j-1,k}(t')| \le C e^{-(\sigma_0+\sigma)t}, $$
+
+which is what was expected.
+
+* Now, from the preceding induction, there exists $\tilde{\sigma}_0 \in (e_j - \sigma, e_j)$ such that
+
+$$ \|E_{j-1}(t)\|_{H^1 \times L^2} \le C e^{-\tilde{\sigma}_0 t}, $$
+
+from which we deduce
+
+$$ |(e^{e_k t} \alpha_{-,j-1,k})'| \le C e^{-(\sigma+\tilde{\sigma}_0-e_k)t}. $$
+
+Now, for $k \in \{1, \dots, j-1\}$, $e_k - \sigma - \tilde{\sigma}_0 \le e_{j-1} - \sigma - e_j < 0$, we thus have
+
+$$ |\alpha_{-,j-1,k}(t)| \le C e^{-(\sigma_0+\tilde{\sigma})t} \le C e^{-e_j t}. $$
+
+For $k=j$, we have $|(e^{e_j t} \alpha_{-,j-1,j})'| \le C e^{(e_j - \tilde{\sigma}_0 - \sigma)t}$. Thus, there exists $A_j \in \mathbb{R}$ such that
+
+$$ e^{e_j t} \alpha_{-,j-1,j}(t) \to A_j, \quad \text{as } t \to +\infty. $$
+
+For $k \in \{j+1, \dots, N\}$, we have $\sigma + \tilde{\sigma}_0 - e_k < \sigma + e_j - e_k < 0$, thus by integration
+
+$$ 
+\begin{aligned}
+|\alpha_{-,j-1,k}(t)| &\le Ce^{-e_k t} + Ce^{-(\tilde{\sigma}_0+\gamma)t} \\
+&\le Ce^{-e_j t}.
+\end{aligned}
+ $$
+
+Hence,
+
+$$ \|E_{j-1}(t)\|_{H^1 \times L^2} \le C \sup_{t' \ge t} \sum_{k=1}^{N} |\alpha_{-,j-1,k}(t')| \le C e^{-e_j t}. $$
+
+Let us conclude the proof of Proposition 4.12. We define at this stage $E_j := U - \Phi_{A_1, \dots, A_j}$. We immediately have
+
+$$ E_j(t) = E_{j-1}(t) + \Phi_{A_1, \dots, A_{j-1}}(t) - \Phi_{A_1, \dots, A_j}(t). $$
+
+Then,
+
+$$ 
+\begin{align*}
+\|E_j(t)\|_{H^1 \times L^2} &\le \|E_{j-1}(t)\|_{H^1 \times L^2} + \|(\Phi_{A_1, \dots, A_j}(t) - \Phi_{A_1, \dots, A_{j-1}}(t)) - A_j e^{-e_j t} Y_{+,j}(t)\|_{H^1 \times L^2} \\
+&\quad + \|A_j e^{-e_j t} Y_{+,j}(t)\|_{H^1 \times L^2} \\
+&\le Ce^{-e_j t}.
+\end{align*}
+ $$

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+What is more,
+
+$$
+\begin{align*}
+\alpha_{-,j,k}(t) &= \langle E_j(t), Z_{-,k}(t) \rangle \\
+&= \alpha_{-,j-1,k}(t) - A_j e^{-e_j t} \langle Y_{+,j}, Z_{-,k} \rangle + O(e^{-(e_j+\sigma)t}).
+\end{align*}
+$$
+
+For $k = 1, \dots, j-1$, we have:
+
+$$
+\begin{align*}
+e^{e_k t} |\alpha_{-,j,k}(t)| &\le C e^{e_k t} |\alpha_{-,j-1,k}(t)| + O(e^{-(e_j-e_k+\sigma)t}) \\
+&\le C e^{(e_k-e_j)t} \xrightarrow{t \to +\infty} 0.
+\end{align*}
+$$
+
+For $k = j$,
+
+$$
+e^{e_j t} \alpha_{-,j,j}(t) = e^{e_j t} \alpha_{-,j-1,j}(t) - A_j + O(e^{-\sigma t}) \xrightarrow{t \to +\infty} 0.
+$$
+
+This finishes the induction argument.
+
+Finally we obtain that $U = \Phi_{A_1, \dots, A_N}$ by means of
+
+**Corollary 4.14.** For *t* sufficiently large, ||*E*<sub>*N*</sub>(*t*)||<sub>H<sup>1</sup>×L<sup>2</sup></sub> = 0.
+
+*Proof.* As in the preceding proofs, the following bounds hold:
+
+$$
+(4.33) \qquad \|E_N(t)\|_{H^1 \times L^2} \le C \sup_{t' \ge t} \sum_{i=1}^N |\alpha_{-,N,i}(t')|
+$$
+
+and
+
+$$
+(4.34) \quad |a'_{-,N,i}(t) + e_i a_{-,N,i}(t)| \le C (e^{-\sigma t} \|E_N(t)\|_{H^1 \times L^2} + \|E_N(t)\|_{H^1 \times L^2}^2).
+$$
+
+We observe that $t \mapsto e^{e_i t} (e^{-\sigma t} \|E_N(t)\|_{H^1 \times L^2} + \|E_N(t)\|_{H^1 \times L^2}^2)$ is integrable on $[t_0, +\infty)$; this is
+due to the fact that $\|E_N(t)\|_{H^1 \times L^2} \le Ce^{-e_N t}$. Since for all $i=1, \dots, N$, $e^{e_i t} \alpha_{-,N,i}(t) \to 0$ as
+$t \to +\infty$, we obtain by integration of (4.34) on $[t, +\infty)$:
+
+$$
+|\alpha_{-,N,i}(t)| \le C \left( e^{-\sigma t} \sup_{t' \ge t} \|E_N(t')\|_{H^1 \times L^2} + \sup_{t' \ge t} \|E_N(t')\|_{H^1 \times L^2}^2 \right).
+$$
+
+Then, using (4.33), we obtain
+
+$$
+\sup_{t' \ge t} \|E_N(t')\|_{H^1 \times L^2} \le C \left( e^{-\sigma t} \sup_{t' \ge t} \|E_N(t')\|_{H^1 \times L^2} + \sup_{t' \ge t} \|E_N(t')\|_{H^1 \times L^2}^2 \right).
+$$
+
+This implies that ||*E*<sub>*N*</sub>(*t*)||<sub>*H*<sup>1</sup>×*L*<sup>2</sup></sub> = 0 for *t* sufficiently large.
+
+5. CONSTRUCTION OF A ONE-PARAMETER FAMILY OF SOLUTIONS CONVERGING TO A SOLITON
+
+The goal of this section is to prove the existence part in Theorem 1.6. Once again we restrict our focus to *d* = 1.
+
+**5.1. Outline of the construction.** Let $A \in \mathbb{R}$.
+
+Let $(S_n)_{n \in \mathbb{N}}$ be an increasing sequence of real numbers which tends to $+\infty$ and, for all $n \in \mathbb{N}$,
+define $u_n$ as the maximal solution of (NLKG) such that
+
+$$
+(5.1) \qquad U_n(S_n) = R_\beta(S_n) + A e^{-e_\beta S_n} Y_{+, \beta}(S_n),
+$$
+
+with obvious notations.
+
+We aim at proving the following key proposition:
+
+**Proposition 5.1.** There exist $t_0 \ge 0$ and $C_0 \ge 0$ such that for *n* large,
+
+$$
+(5.2) \quad \forall t \in [t_0, S_n], \quad \|U_n(t) - R_\beta(t) - A e^{-e_\beta t} Y_{+, \beta}(t)\|_{H^1 \times L^2} \le C_0 e^{-2e_\beta t}.
+$$

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+To this end, we will set up a bootstrap argument and show
+
+**Proposition 5.2.** There exist $\alpha_0 > 0$, $C_0 > 0$, and $t_0 \ge 0$ such that for $n$ sufficiently large, if there exists $t_n^* \in [t_0, S_n]$ such that for all $t \in [t_n^*, S_n]$,
+
+$$ (5.3) \qquad \|U_n(t) - R_\beta(t) - A e^{-e_\beta t} Y_{+,\beta}(t)\|_{H^1 \times L^2} \le \alpha_0, $$
+
+then for all $t \in [t_n^*, S_n]$,
+
+$$ (5.4) \qquad \|U_n(t) - R_\beta(t) - A e^{-e_\beta t} Y_{+,\beta}(t)\|_{H^1 \times L^2} \le C_0 e^{-2e_\beta t}. $$
+
+Let us show how to deduce Proposition 5.1 from Proposition 5.2.
+
+*Proof of Proposition 5.1.* Assume momentarily that Proposition 5.2 holds true. Let us consider $\alpha_0$ and $C_0$ as in Proposition 5.2 and suppose (even if it means enlarging $t_0$) that $C_0 e^{-2e_\beta t_0} \le \frac{\alpha_0}{2}$. We define for all $n$ such that $S_n > t_0$:
+
+$$ t_n^* := \inf \{ t \in [t_0, S_n], \forall \tau \in [t, S_n], \|U_n(\tau) - R_\beta(\tau) - A e^{-e_\beta \tau} Y_{+, \beta}(\tau)\|_{H^1 \times L^2} \le \alpha_0 \}. $$
+
+By (5.1) and by continuity in time of $U_n$, $R_\beta$, and $Y_{+,\beta}$, $t_n^*$ is indeed well-defined and we necessarily have $t_0 \le t_n^* < S_n$. Since (5.3) implies (5.4), for all $t \in [t_n^*, S_n]$,
+
+$$ \begin{align*} \|U_n(t) - R_\beta(t) - A e^{-e_\beta t} Y_{+, \beta}(t)\|_{H^1 \times L^2} &\le C_0 e^{-2e_\beta t} \\ &\le C_0 e^{-2e_\beta t_0} \\ &\le \frac{\alpha_0}{2}. \end{align*} $$
+
+Let us assume for the sake of contradiction that $t_n^* > t_0$ for some $n$. Then, observing the preceding inequality, we obtain (again by continuity in time of $U_n$, $R_\beta$, and $Y_{+,\beta}$) the existence of $\tau_n > 0$ such that $t_n^* - \tau_n \ge t_0$ and for all $t \in [t_n^* - \tau_n, S_n]$,
+
+$$ \|U_n(t) - R_\beta(t) - A e^{-e_\beta t} Y_{+, \beta}(t)\|_{H^1 \times L^2} \le \frac{3\alpha_0}{4} < \alpha_0. $$
+
+This contradicts the definition of $t_n^*$ as an infimum. Hence $t_n^* = t_0$ and (5.3) (and thus (5.4)) holds on $[t_0, S_n]$ for all $n$. This achieves the proof of Proposition 5.1. □
+
+The existence of $u^A$ (and $U^A$), as stated in Theorem 1.6 is a consequence of Proposition 5.1 and the continuity of the flow of (NLKG) for the weak $H^1 \times L^2$ topology. We will not detail the construction of $U^A$ considering that it is a sort of cut and paste of what was done in order to prove Proposition 3.1 in the context of multiple solitons.
+
+Similarly, we do not repeat the arguments exposed at the beginning of section 3 which justify that the map $A \mapsto u^A$ is one-to-one. We devote the next subsection to the proof of Proposition 5.2.
+
+**5.2. Proof of Proposition 5.2.** We assume that $U_n(t)$ is defined on some interval $[t_n^*, S_n]$ and satisfies (5.3). We want to show that (5.4) holds, provided that the parameters $\alpha_0$ and $t_0$ are well chosen.
+
+In this subsection again, for notation purposes and ease of reading, we sometimes omit the index $n$ and also write $O(G(t))$ in order to refer to a function $g$ which a priori depends on $n$ and such that there exists $C \ge 0$ (independent of $n$) such that for all $n$ large and for all $t \in [t_n^*, S_n]$, $|g(t)| \le C|G(t)|$.

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+**5.2.1. Step 1: Set up of a modulation argument.**
+
+**Lemma 5.3.** For $t_0 \ge 0$ sufficiently large and $\alpha_0 > 0$ sufficiently small, there exists a unique $\mathscr{C}^1$
+function $x : [t_n^*, S_n] \to \mathbb{R}$ such that if we set
+
+$$
+W_n(t) := U_n(t) - \tilde{R}_\beta(t) - Ae^{-e_\beta t} \tilde{Y}_{+\beta}(t),
+$$
+
+with $\tilde{R}_\beta(t) := R_\beta(t, \cdot -x(t))$ and $\tilde{Y}_{+\beta}(t) := Y_{+\beta}(t, \cdot -x(t))$, then for all $t \in [t_n^*, S_n]$,
+
+$$
+(5.5) \qquad \langle W_n(t), \partial_x \tilde{R}_\beta(t) \rangle = 0.
+$$
+
+Moreover there exists $K_1 > 0$ such that for all $t \in [t_n^*, S_n]$,
+
+$$
+(5.6) \qquad \|W_n(t)\|_{H^{1}\times L^2} + |x(t)| \le K_1 \alpha_0,
+$$
+
+$$
+(5.7) \qquad |\dot{x}(t)| \le K_1 (\|W_n(t)\|_{H^1 \times L^2} + e^{-2e_\beta t}).
+$$
+
+*Remark 5.4.* Notice that by uniqueness of the function *x* and by definition of *u*<sub>*n*</sub> (see (5.1)), we have *W*<sub>*n*</sub>(*S*<sub>*n*</sub>) = 0 and *x*(*S*<sub>*n*</sub>) = 0.
+
+*Proof.* The existence of *x* such that (5.5) is granted and the existence of *K*₂ > 0 such that
+
+$$
+\|W_n(t)\|_{H^1 \times L^2} + |x(t)| \leq K_2 \alpha_0
+$$
+
+are standard consequences of the implicit function theorem.
+
+Now, let us prove (5.7). For this, we notice that $W = W_n$ satisfies the following equation:
+
+$$
+(5.8) \quad \partial_t W = \begin{pmatrix} 0 & Id \\ \partial_x^2 - Id + f'(Q_\beta) & 0 \end{pmatrix} (W + A e^{-e_\beta t} \tilde{Y}_{+\beta}) + \left( f(u) - f(Q_\beta) - f'(Q_\beta)(u - Q_\beta) \right) \\
+\qquad + \dot{x}(t) \partial_x \tilde{R}_\beta - A e^{-e_\beta t} (\partial_t \tilde{Y}_{+\beta} - \dot{x}(t) \partial_x \tilde{Y}_{+\beta} - e_\beta \tilde{Y}_{+\beta}),
+$$
+
+where $\tilde{Q}_{\beta}(t,x) = Q_{\beta}(t,x-x(t))$.
+
+Since $\frac{d}{dt}\langle W, \partial_x \tilde{R}_\beta \rangle = 0$, we have:
+
+$$
+\langle \partial_t W, \partial_x \tilde{R}_\beta \rangle + \langle W, \partial_{tx} \tilde{R}_\beta - \dot{x} \partial_x^2 \tilde{R}_\beta \rangle = 0.
+$$
+
+Observing moreover that
+
+$$
+f(u) - f(\tilde{Q}_{\beta}) - f'(\tilde{Q}_{\beta})(u - \tilde{Q}_{\beta}) = O\left(\|U - \tilde{R}_{\beta}\|_{H^{1}\times L^{2}}^{2}\right) = O\left(\|W(t)\|_{H^{1}\times L^{2}}^{2} + e^{-2e_{\beta}t}\right)
+$$
+
+by Taylor formula ($f$ is $\mathcal{C}^2$), we have thus:
+
+$$
+\begin{align*}
+0 &= \left\langle \begin{pmatrix} 0 & Id \\ \partial_x^2 - Id + f'(\tilde{Q}_\beta) & 0 \end{pmatrix} W, \partial_x \tilde{R}_\beta \right\rangle + O\left(\|W(t)\|_{H^{1}\times L^2}^2 + e^{-2e_\beta t}\right) + \dot{x}(t) \| \partial_x R_\beta \|_{H^{1}\times L^2}^2 \\
+&\quad + A e^{-e_\beta t} \left\langle (\beta + \dot{x}) \partial_x \tilde{Y}_{+\beta} + e_\beta \tilde{Y}_{+\beta}, \partial_x \tilde{R}_\beta \right\rangle - (\beta + \dot{x}) \langle W, \partial_x^2 \tilde{R}_\beta \rangle \\
+&= \dot{x} \left( \| \partial_x R_\beta \|_{H^{1}\times L^2}^2 + A e^{-e_\beta t} \langle \partial_x \tilde{Y}_{+\beta}, \partial_x \tilde{R}_\beta \rangle - \langle W, \partial_x^2 \tilde{R}_\beta \rangle \right) + O\left(\|W(t)\|_{H^{1}\times L^2}^2 + e^{-2e_\beta t}\right) \\
+&\quad + \left\langle \begin{pmatrix} \beta \partial_x & Id \\ \partial_x^2 - Id + f'(\tilde{Q}_\beta) & \beta \partial_x \end{pmatrix} W, \partial_x \tilde{R}_\beta \right\rangle + A e^{-e_\beta t} \left\langle \begin{pmatrix} \beta \partial_x & Id \\ \partial_x^2 - Id + f'(\tilde{Q}_\beta) & \beta \partial_x \end{pmatrix} \tilde{Y}_{+\beta}, \partial_x \tilde{R}_\beta \right\rangle.
+\end{align*}
+$$
+
+Notice that we have used $\langle Y_{+\beta}, \partial_x R_\beta \rangle = 0$ (see Proposition 2.1). We now observe that
+
+$$
+\left( \begin{pmatrix} \beta & 0 \\ 0 & Id \end{pmatrix} - Id + f'(Q_{\beta}) & 0 \\ 0 & Id \end{pmatrix} = -J\tilde{H}_{\beta},
+$$
+
+where $\tilde{H}_{\beta}$ is the matrix operator defined like $H_{\beta}$ by replacing $Q_{\beta}$ by $\tilde{Q}_{\beta}$, that is
+
+$$
+(5.9) \qquad \tilde{H}_{\beta} := \begin{pmatrix} -\partial_x^2 + Id - f'(\tilde{Q}_{\beta}) & -\beta\partial_x \\ 0 & \beta\partial_x \\ 0 & Id \end{pmatrix}.
+$$
+
+We obtain:
+
+$$
+\left\langle\left(\begin{array}{cc} 0 & 0 \\ 0 & Id \\ 0 & 0 \end{array}\right)-Id+f^{\prime}\left(\tilde{Q}_{\beta}\right)\right\rangle=\left\langle\tilde{Y}_{+,\beta}, \tilde{H}_{\beta} J \partial_{x} \tilde{R}_{\beta}\right\rangle=\left\langle\tilde{Y}_{+,\beta}, 0\right\rangle=0.
+$$