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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
f3d9c6215f8a5e9c1b86fbd7164ab05632e7ae2f | subsection | 32 | 76 | Canonical form of equation for | Hence by (REF ) {\rm Re{\hspace{1.42262pt}}}Z^{\prime }_{21} reduces to{\rm Re{\hspace{1.42262pt}}}Z^{\prime }_{21}=-{\rm Re{\hspace{1.42262pt}}}2\frac{\langle ({\bf C}_T\!-2i\mu _T-0)^{-1}{\bf P}_T^c
jE_2[u,u], E_2[u,u]\rangle }{\varkappa }
=-\frac{2}{\delta }{\rm Im{\hspace{1.42262pt}}}\langle R(2i\mu +0){\bf P}_T^c
... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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614975c79993a3e4c3fda745096ebc9c2345df61 | subsection | 33 | 76 | Canonical form of equation for | Hence by the Cauchy residue theorem we have\langle R(2i\mu +0)\alpha , j\alpha \rangle =
-\frac{1}{2\pi i}\int \limits _{{\cal C}_+\cup {\cal C}_-}d\lambda ~
\frac{\langle (R(\lambda +0)-R(\lambda -0))\alpha , j\alpha \rangle }{\lambda -2i\mu -0}Now we use the representationR(\lambda +0)-R(\lambda -0)=-\frac{\tau _{\pm... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
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72305ec6341d0863a3a65e12efba6c1a6f232e3b | subsection | 34 | 76 | Canonical form of equation for | Using that
\frac{1}{\nu +i0}=p.v.\frac{1}{\nu }-i\pi \delta (\nu )
where p.v. is the Cauchy principal value, we have\langle R(2i\mu +0)\alpha , j\alpha \rangle \!\!\!&=&\!\!\!-\frac{1}{16\pi }\int _{-\infty }^{-\omega }\;\frac{d\nu }{k_-|D|^2}
\frac{\langle \tau _-,j\alpha \rangle \overline{\langle \tau _-,j\alpha \r... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
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a7e1a23679781f24885aa12302e463825a463d75 | subsection | 35 | 76 | Canonical form of equation for | Now we estimate the remainder \tilde{Z}_R.Lemma 5.4
The remainder \tilde{Z}_R has the form\tilde{Z}_R={\cal R}_1(\omega ,|z|+\Vert f\Vert _{L^{\infty }_{-\beta }})
\Bigl [(|z|^2+\Vert f\Vert _{L^{\infty }_{-\beta }})^2+
|z||\omega _T-\omega |\Vert h\Vert _{L^{\infty }_{-\beta }}
+|z|\Vert k_1\Vert _{L^{\infty }_{-\bet... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
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9da113dc8936468418a9391af0b957a5da763386 | subsection | 36 | 76 | Canonical form of equation for | We can apply now the method of normal coordinates to equation (REF ).Lemma 5.5 (cf. )There exist coefficients c_{ij} such that the new function z_1 defined byz_1=z+c_{20}z^2+c_{11}z\overline{z}+c_{02}\overline{z}^2+c_{30}z^3
+c_{12}z\overline{z}^2+c_{03}\overline{z}^3,satisfies an equation of the form
\dot{z}_1=i\mu (\... | {
"cite_spans": [
{
"arxiv_id": "",
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"end": 517,
"openalex_id": "",
"raw": "V.S. Buslaev, C. Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations,Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20 (2003), no.3, 419-475.",
"source_ref_id":... | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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15467de4bf2d1252a64a09ed73eef4af75504d50 | subsection | 37 | 76 | Canonical form of equation for | The equation satisfied by y is simply obtained by multiplying (REF ) by \overline{z}_1
and taking the real part:\dot{y}=2{\rm Re{\hspace{1.42262pt}}}(iK_T)y^2 +Y_R,where|Y_R|\!=\!{\cal R}_1(\omega ,|z|+\Vert f\Vert _{L^{\infty }_{-\beta }})|z|
\Bigl [(|z|^2+\Vert f\Vert _{L^{\infty }_{-\beta }})^2+
|z||\omega _T-\omega... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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2ce7c97da7f41b88d070cb556a556ace87e26fc3 | subsection | 38 | 76 | Bound for | Now we obtain a uniform bound for |\omega _T-\omega (t)| on the interval [0,T].&&\!\!\!\!\!\!\!\!
|\omega _T-\omega (t)|\le |\omega _{1T}-\omega _1(t)|+|\omega _{1T}-\omega _T|+|\omega _1(t)-\omega (t)|\\
&\!\!\!\!\!\le &\!\!\!\!\int _t^T|\dot{\omega }_1(\tau )|d\tau +
{\cal R}(\omega _T,|z_T|+\Vert f_T\Vert _{L^{\inft... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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33a98dac107a17bfc0c2ba592bab76862622a59a | subsection | 39 | 76 | Bound for | Then|\omega _T-\omega |\le {\cal R}_2(\omega , |z|+\Vert f\Vert _{L^{\infty }_{-\beta }})
\Bigl [\int _t^T\!(|z|^2+\Vert f\Vert _{L^{\infty }_{-\beta }})^2d\tau +
(|z_T|+\Vert f_T\Vert _{L^{\infty }_{-\beta }})^2
+(|z|+\Vert f\Vert _{L^{\infty }_{-\beta }})^2\Bigr ]As in (REF ),
we suppose the smallness condition:|z(0)... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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655b0593659d3bb7d30428e5dd60a986f00dca67 | subsection | 40 | 76 | Bound for | Thereforey(0)=|z_1(0)|^2\le \varepsilon +{\cal R}(\omega ,|z(0)|)\varepsilon ^{3/2}.From the formula h={\bf P}_T^cf=f+({\bf P}^d-{\bf P}_T^d)f, we see that\Vert h(0)\Vert _{L^1_\beta }\le c\varepsilon ^{3/2}+{\cal R}_1(\omega )|\omega _T-\omega |
\Vert f(0)\Vert _{L^{\infty }_{-\beta }}.Lemma 5.7
The function k_1 defi... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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767054d08a9e5554c5e0faf7a884536c77fd9c0e | subsection | 41 | 76 | Bound for | Therefore, the bounds (), (), and assumption (REF )
imply (REF ). | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
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1df695673fb267825c5e8f5571781c47b35f999f | subsection | 42 | 76 | Large time asymptotics | In this section we will make use of the dispersive estimates
given in § to prove the asymptotic representation
for the solution of (REF )
with initial data as in Theorem REF . | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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4224e64f28bfa22ea3d0ae362e4bda28d73ac632 | subsection | 43 | 76 | Definition of majorants | We define the quantities{\mathbb {M}}_0(T)&=&\max \limits _{0\le t\le T}|\omega _T-\omega |\Big (\frac{\varepsilon }{1+\varepsilon t}\Big )^{-1}\\
{\mathbb {M}}_1(T)&=&\max \limits _{0\le t\le T}|z(t)|\Big (\frac{\varepsilon }{1+\varepsilon t}\Big )^{-1/2}\\
{\mathbb {M}}_2(T)&=&\max \limits _{0\le t\le T}\Vert h_1\V... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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f6d2bc559aea7f8c45718f155b561f6945373098 | subsection | 44 | 76 | Estimates of remainders and initial data | Lemma 6.1
The remainder Y_R defined in (REF ) satisfies the estimate|Y_R|={\cal R}(\varepsilon ^{1/2}{\mathbb {M}})\frac{\varepsilon ^{5/2}}{(1+\varepsilon t)^2\sqrt{\varepsilon t}}
(1+|{\mathbb {M}}|)^5.Using the equality f=g+h=g+k+k_1+h_1, Lemma REF and the
definitions of the {\mathbb {M}}_j,
the remainder Y_R is bo... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
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0.016387939453125,
0.1435546875,
0.214111328125,
0.242431640625,
0.1168212890625,
0.256103515625,
0.... | |
a5d18c6105e1072bc21ca22147b47a12c893aab9 | subsection | 45 | 76 | Estimates of remainders and initial data | \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
+\Big (\frac{\varepsilon }{1+\varepsilon t}\Big )^{3/2}{\mathbb {M}}_0{\mathbb {M}}_1
\Big (\frac{\varepsilon }{1+\varepsilon t}{\mathbb {M}}_1^2+\frac{\varepsilon }{(1+ t)^{3/2}}
+\Big (\frac{\varepsilon }{1+\varepsilon t}\Big )^{3/2}{\mathbb {M}}_3\... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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61878692f972380d8cc6e7ae2d4651e92cc040b0 | subsection | 46 | 76 | Estimates of remainders and initial data | For the A_m we now obtain:Lemma 6.3\Vert A_{m}\Vert _{{\cal M}_{\beta }}= {\cal R}(\varepsilon ^{1/2}{\mathbb {M}})
\Big (\frac{\varepsilon }{1+\varepsilon t}\Big )^{3/2}\Big ({\mathbb {M}}_1^3+\varepsilon ^{1/2}(1+|{\mathbb {M}}|)^3\Big ).Estimate (REF ) implies\Vert A_{m}\Vert _{{\cal M}_{\beta }}
={\cal R}_2(\omega ... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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31811e76aadc3e593dcec430c89f117266340413 | subsection | 47 | 76 | Integral inequalities and decay in time | This section is devoted to a study of the system:\dot{y}=2{\rm Re{\hspace{1.42262pt}}}(iK_T)y^2+Y(t),\dot{h}_1={\bf C}_Mh_1+H(x,t),under some assumptions on the initial data, and on the inhomogeneous
(or source) terms Y and H. Equation (REF ) for y is of Ricatti type.
For the initial data, we assumey(0)\le \varepsilon ... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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331403edcf6c58c0d69052fa10d97f270631976d | subsection | 48 | 76 | Integral inequalities and decay in time | Finally, corresponding to (REF ), we work under the
assumption {\rm Re{\hspace{1.42262pt}}}(iK_T)=-{\rm Im{\hspace{1.42262pt}}}K_T<0.Lemma 6.4 ()
The solution of (REF ), with initial condition and source term
satisfying (REF )
and (REF ) respectively, is bounded as follows for t>0:|y(t)-\frac{y(0)}{1+2{\rm Im{\hspace{1... | {
"cite_spans": [
{
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"end": 465,
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"raw": "V.S. Buslaev, C. Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations,Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20 (2003), no.3, 419-475.",
"source_ref_id":... | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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7c0d1dbdc33f8cae056e0c3b51c2ed3b6c23fa81 | subsection | 49 | 76 | Inequalities for majorants | In this section we estimate in turn the three majorants
{\mathbb {M}}_0,{\mathbb {M}}_1,{\mathbb {M}}_2.Lemma 6.6
The majorants {\mathbb {M}}_0(T), {\mathbb {M}}_1(T), and {\mathbb {M}}_2(T) satisfy{\mathbb {M}}_0(T)={\cal R}(\varepsilon ^{1/2}{\mathbb {M}})\Bigl [(1+{\mathbb {M}}_1)^4+\varepsilon (1+|{\mathbb {M}}|)^... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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f7321cf1da658b97608034a7532020c4ef589542 | subsection | 50 | 76 | Inequalities for majorants | Using (REF ) as well as (REF ) to bound the initial condition y(0),
it follows that
y\le {\cal R}(\varepsilon ^{1/2}{\mathbb {M}})\Big [\frac{\varepsilon }{1+\varepsilon t}+\Big (\frac{\varepsilon }{1+\varepsilon t}\Big )
^{3/2}(1+|{\mathbb {M}}|)^5\Big ].
Therefore|z|^2\le y+ {\cal R}(\omega )|z|^3
\le {\cal R}(\var... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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d15842706e32fbea168bf092efa5355216ce2297 | subsection | 51 | 76 | Uniform bounds for majorants | Now we prove that if \varepsilon is sufficiently small, all the {\mathbb {M}}_i
are bounded uniformly in T and \varepsilon .Lemma 6.7
For \varepsilon sufficiently small, there exists a constant M independent of T and \varepsilon ,
such that,|{\mathbb {M}}(T)|\le M.Combining the inequalities (REF )-(REF ) for the {\mat... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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9a670030a1f5d963549101d313fee2cdb54c190f | subsection | 52 | 76 | Large time behaviour of solution | Here we deduce from corollary REF a theorem
which describe a large time behaviour of the solution.
Notice that in the decomposition f=g+h=g+h_1+k+k_1, a fixed time T
has been chosen, and all the components depend on \omega (T). From the above proposition,
we know that \omega (t) has a limit \omega _{+} as t\rightarrow ... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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f1f6333b340cf547b96a470e4354668e1a270c55 | subsection | 53 | 76 | Large time behaviour of solution | A corresponding
statement also holds for t\rightarrow -\infty . | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
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5a24c8c54fc53496cda0fa455d6cc75385ee99c8 | subsection | 54 | 76 | Scattering asymptotics | In this section we obtain the scattering asymptotics (REF ). | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
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17d3b4f2bab9570f6ff327a16dd32eded0c02a31 | subsection | 55 | 76 | Large time behavior of | We start with equation (REF ) for z_1, rewritten as
\dot{z}_1=i\mu z_1+iK_{+}|z_1|^2z_1+\widehat{\widehat{Z}}_R
with K_{+}=K(\omega _{+}). By (REF ) the inhomogeneous term
\widehat{\widehat{Z}}_R
satisfies the estimate|\widehat{\widehat{Z}}_R|
={\cal R}(\varepsilon ^{1/2}M)\frac{\varepsilon ^2}{(1+\varepsilon t)^{3/2... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
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32032ac59c3f2bdef49bc1588abd35b182cd7780 | subsection | 56 | 76 | Large time behavior of | The solution z_1 of (REF ) is written in the formz_1=\!\frac{e^{i\int _0^t\mu (t_1)dt_1}}{(1+{\epsilon }k_{+}t)^{\frac{1}{2}(1-i\delta )}}
\Big [z_1(0)+\!\int _0^t\!\!e^{-i\int \limits _0^s\mu (t_1)dt_1}
(1+{\epsilon }k_{+}s)^{\frac{1}{2}(1-i\delta )}Z_1(s)ds\Big ]
=z_{\infty }\frac{e^{i\int _0^t\mu (t_1)dt_1}}{(1+{\ep... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
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819649050d3448dc015254f3c115a284afd491d6 | subsection | 57 | 76 | Large time behavior of | Therefore z(t)=z_1(t)+ O(t^{-1}) satisfiesz(t)= z_{+}\frac{e^{i\int _0^t\mu (t_1)dt_1}}{(1+{\epsilon }k_{+}t)^{\frac{1}{2}(1-i\delta )}}
+ O(t^{-1}),\;t\rightarrow \infty ,\quad z_{+}=z_{\infty }(\omega _{+}).From these formulas for z(t), the
asymptotic behavior of \omega (t) and \gamma (t) can be
deduced as in , leadi... | {
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{
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"raw": "V.S. Buslaev, C. Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations,Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20 (2003), no.3, 419-475.",
"source_ref_id":... | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
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43f13837d140e95f713812ed1d6bb5c5df61206b | subsection | 58 | 76 | Soliton asymptotics | Here we prove the statement (REF ) in
our main theorem REF . | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
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] | |
e7974fb056b9b9989049e12aedf83518c809b1ef | subsection | 59 | 76 | Soliton asymptotics | To achieve this
we look for the solution \psi (x,t) to (REF ), in the
corresponding complex form \psi =s+{\rm v}+f,
wheres(x,t)=\psi _{\omega (t)}(x)e^{i\theta (t)},\quad \dot{\theta }(t)=\omega (t)+\dot{\gamma }(t)is the accompanying soliton, and{\rm v}(x,t)=v(x,t)e^{i\theta (t)},\quad v(x,t)=\big (z(t)+\overline{z}(t... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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4e19df2d4479bb409a2b0b996952a57599c5ebcf | subsection | 60 | 76 | Soliton asymptotics | The function f(x,t) which is a solution of (REF ) can be expressed asf(t)&= &W(t)f(0)+\int _0^t W(t-\tau )R(\tau )
d\tau \\
&= &W(t)\Big (f(0)+
\int _0^{\infty } W(-\tau )R(\tau )d\tau \Big )
-\int ^{\infty }_t W(t-\tau )R(\tau )d\tau = W(t)\phi _{+}+r_{+}(t)where W(t) is the dynamical group of the free Schrödinger equ... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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c7fdd623d13e29df606e83334ff88c16c123951c | subsection | 61 | 76 | Soliton asymptotics | To prove the L^2-properties, let us change the variable to \tau =1/u to get:\psi (x):=\frac{1}{\sqrt{-4\pi i}}\int _0^\infty e^{-iux^2/4}~\eta (u)~du
=\frac{1}{\sqrt{-2i}}{\mathcal {F}}_{u\rightarrow x^2/4}(\theta (u)\eta (u)),
\qquad \eta (u)=\Pi (1/u)/u^{3/2},where
{\mathcal {F}}_{u\rightarrow \xi }(f(u))=\hat{f}(\xi... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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fc689ccffbb9ed2aec92f5a2d5661f52eb530611 | subsection | 62 | 76 | Soliton asymptotics | The Young inequality then implies that\Vert \rho (x,t)\Vert _{L^2}\le \Vert \rho (x,t)(1+|x|)^{1/q}\Vert _{L^{q}}
\Vert (1+|x|)^{-1/q}\Vert _{L^{r}}\le Ct^{-\frac{1-p/2}{p}},\;
q^{-1}+r^{-1}=1/2if r>q.
To have r>q, we must take q<4, or
equivalently p>4/3. Hence, we have \nu =(1-p/2)/p<1/4.
The second lemma studies
the... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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5b2f23ac5cc91b8f3f6fd6dcfffc51f35ec38f62 | subsection | 63 | 76 | Soliton asymptotics | Then\int _0^{\infty }\Pi (\tau ) W(-\tau )\psi d\tau \in C_b({\mathbb {R}})\cap L^2({\mathbb {R}})and\bigl \Vert \int ^{\infty }_t\Pi (\tau ) W(t-\tau )\psi d\tau \bigr \Vert _{C_b({\mathbb {R}})\cap L^2({\mathbb {R}})}
\le C t^{-1/3},\quad t>1.Since \Vert W(t)\psi \Vert _{C_b}= O(t^{-1/2}) then C_b– properties are evi... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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857f11626e3624c368a631235ee2477a3e00664f | subsection | 64 | 76 | Soliton asymptotics | It suffices to prove thatI(t)=\Big \Vert \int _t^{\infty }\frac{e^{i(\xi ^2+\omega _+-2\mu _+)\tau }
\hat{\psi }(\xi )~d\tau }{1+\tau }\Big \Vert _{L^2}={\cal O}(t^{-1/3})For the fixed 0<\beta <1 let us define\chi _{\tau }(\xi )=\left\lbrace \begin{array}{ll}1,~{\rm if}~|\xi -\sqrt{2\mu _+-\omega _+}|\le 1/\tau ^{\beta... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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ee17e77ca0e7b1943efd210631265179720a72ae | subsection | 65 | 76 | Soliton asymptotics | Hence (REF ) follows.Remark 7.5
The t\rightarrow -\infty case is handled in an identical way. | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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88a52322b307e1ef121f13fd644123e8fc376275 | subsection | 66 | 76 | Eigenfunctions of discrete spectrum | Here we find the function u=u(\omega ) satisfying
{\bf C}u=\lambda u, where \lambda =i\mu .
Using the definition of the operator {\bf C}, we obtain\left(
\begin{array}{rcr}-\lambda &&-\Delta +\omega \\
\Delta -\omega &&-\lambda \end{array}\right)u =
\delta (x)
\left(
\!\begin{array}{cc}
0 &a\\
-a-b &0
\end{array}\!\rig... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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acdf9e232c241120bf48ba72acd6b2de315fee9f | subsection | 67 | 76 | Eigenfunctions of discrete spectrum | Then
{\rm Im{\hspace{1.42262pt}}}k_\pm (\lambda )>0 for \lambda \in {\cal C}_\pm
and we have two corresponding vectors v_\pm =(1,\pm i)
and four linearly independent exponential solutionsv_+e^{\pm ik_+ x}=(1,\, i)e^{\pm ik_+ x},~~~~~~~~~~~~
v_-e^{\pm ik_- x}=(1, -i)e^{\pm ik_- x}.Now we find the solution to (REF ) in ... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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ade1df87986115ea60bbe12dea77b99b5527697e | subsection | 68 | 76 | Eigenfunctions of continuous spectrum | Let \lambda =i\nu with some \nu >\omega .
First, we find an even solution u=\tau _+ to
(REF ) in the form\tau _+=(Ae^{ik_+ |x|}+Be^{-ik_+ |x|})v_{+}+Ce^{ik_- |x|}v_{-}Similarly (REF ) and (REF ), we obtain\left\lbrace
\begin{array}{l}
2ik_{+}(A-B)=-\alpha (A+B)-\beta C
\\
\\
2ik_{-}C=-\beta (A+B)-\alpha C
\end{array}\... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
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5fef49f1a12bb30bc434aa521df07377284b3af7 | subsection | 69 | 76 | Proof of Proposition | Denote by B a Banach space with the norm \Vert \cdot \Vert .Lemma C.1 (cf.)
Let the operator K(t), t>0, satisfiesK(t)=\int \zeta (\nu )e^{i\nu t}Q(\nu )d\nu ,\quad Q(\nu ):=\frac{L(\nu )-L(\nu _0)}{\nu -\nu _0},where \zeta \in C_0^\infty ({\mathbb {R}}), \zeta (\nu )=1 in the some vicinity of \nu _0,
and for k=0,1,2M_k... | {
"cite_spans": [
{
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"doi": "",
"end": 403,
"openalex_id": "",
"raw": "A. Komech, E.Kopylova, On Asymptotic stability of moving kink for relativistic Ginsburg-Landau equation, accepted in Commun. Math. Phys., ArXiv:0910.5538",
"source_ref_id": "23d254201e69c1f65b... | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
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765e332a526c6ba617414e0b3d96a4f68136fdb4 | subsection | 70 | 76 | Proof of Proposition | Then we obtaine^{{\bf C}t}({\bf C}-2i\mu -0)^{-1}=-\frac{1}{2\pi i}\int _{-i\infty }^{i\infty }
e^{\lambda t}\frac{R(\lambda +0)- R(2i\mu +0)}{\lambda -2i\mu }~d\lambda=-\frac{1}{2\pi i}\!\int _{-i\infty }^{i\infty }\!
e^{\lambda t}\zeta (\lambda )\frac{R(\lambda +0)- R(2i\mu +0)}{\lambda -2i\mu }~d\lambda -\frac{1}{2\... | {
"cite_spans": [
{
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"raw": "V.Buslaev, A.Komech, E.Kopylova, D.Stuart, On asymptotic stability of solitary waves in a nonlinear Schrödinger equation, Comm. Partial Diff. Eqns 33 (2008), no. 4, 669-705.",
"source_ref_id"... | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
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d0460c98d2031a4dce6a79ff3e8f1c259a0cc8d9 | subsection | 71 | 76 | Proof of Proposition | At the points \lambda =0 and \lambda =\pm i\mu the
integrand has the poles of finite order. Hoverever, all the Laurent
coefficients vanish when applied to {\bf P}^{c}h. Hence for {\bf K}_3(t)
we obtain, twice integrating by parts,\Vert {\bf K}_3(t){\bf P}^ch\Vert _{L^{\infty }_{-\beta }}
\le c(1+t)^{-3/2}\Vert h\Vert _... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
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"E. A. Kopylova",
"D. Stuart"
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1cedbc8f2e2d2cabfadb721033d6adb14b61bb8c | subsection | 72 | 76 | Proof of Lemma | We will use the following representation (see ):{\bf P}^c=\frac{1}{2\pi i}\int _{{\cal C}_{+}}\!({\bf R}(\lambda +0)
-{\bf R}(\lambda -0))d\lambda +\frac{1}{2\pi i}\int _{{\cal C}_{-}}\!({\bf R}(\lambda +0)\!-\!{\bf R}(\lambda -0))d\lambda =
{\bf \Pi }^++{\bf \Pi }^-.Let us decompose the resolvent, as given in (REF ) a... | {
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18e25570108e0a22a43d1151986f4a30cfe96005 | subsection | 73 | 76 | Proof of Lemma | For \lambda \in {\cal C}_+ we have:
k_+ is real, and k_+(\lambda +0)=-k_+(\lambda -0)
while k_- is pure imaginary with
{\rm Im{\hspace{1.42262pt}}}k_->0 and k_-(\lambda +0)=k_-(\lambda -0).Since A_5(\lambda ,x,y) for \lambda \in {\cal C}_+ exponentially
decay if |x|,\,|y|\rightarrow \infty and smallest exponential rate... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
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d5e9ed09308fbd56a137fd3b2c2ba66c94c1d9ea | subsection | 74 | 76 | Fermi Golden Rule | In this section we show that condition (REF ) holds generically in a certain
sense: in particular, if a(\cdot ) is a polynomial function then, generically,
the set of values of C for which (REF ) fails is isolated.
By (REF )\tau _{+}(2i\mu )\mid _{x=0}=(\overline{D}-D)v_{+}+4\beta ik_{+}v_{-}
=-4ik_{+}(\alpha +2ik_{-})... | {
"cite_spans": []
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0d0aa74a6861ee446a4e2680f336dda152640fee | subsection | 75 | 76 | Fermi Golden Rule | Therefore,
(u(0),u(0))=(\rho +1)^2-(\rho -1)^2=4\rho and then\tilde{E}_2[u(0),u(0)]=a^{\prime }(C^2)4\rho \left(\begin{array}{c} C\\0\end{array}\right)
+2a^{\prime \prime }(C^2)C^2(\rho +1)^2\left(\begin{array}{c} C\\0
\end{array}\right)
+2a^{\prime }(C^2)C(\rho +1)\left(\begin{array}{c} \rho +1\\i(\rho -1)
\end{array}... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
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"D. Stuart"
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bfd93e7af38b83a669fcefeec4a22e1e045e09cf | abstract | 0 | 11 | Abstract | We study a scalar phi field that unifies inflation and dark energy with a
long period of a hot decelerating universe in between these two stages of
inflation. A key feature is that the transition between the intermediate
decelerated phase to the dark energy phase is related to a quantum regeneration
of the scalar field... | {
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} | 0807.1880 | Inflation-Dark Energy unified through Quantum Regeneration | [
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e6410eb1ec9bbfed59848fd4cb03e1accce85ea6 | subsection | 1 | 11 | Body | Inflation-Dark Energy unified through Quantum RegenerationA. de la Macorra and F. Briscese
Instituto de Física, Universidad Nacional Autonoma de Mexico, Apdo. Postal 20-364,
01000 México D.F., MéxicoPart of the Collaboration Instituto Avanzado de CosmologiaWe study a scalar \phi field that unifies inflation and dark
e... | {
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11799e800aca216d812e7b9ca9d60628e41bcc11 | subsection | 2 | 11 | Body | If the inflaton decay is not complete
then the remaining energy density of the inflaton redshifts as
matter at late times. The amount of residual energy density
must be fine tuned if one wants to be interpreted as dark matter.
However, in our case the uniton can no longer have a
minimum at a finite value for \phi since... | {
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"raw": "A. de la Macorra and G. Piccinelli, Phys. Rev. D 61, 123503 (2000), arXiv:hep-ph/9909459; A. de la Macorra, C. Stephan-Otto, Phys. Rev. D 65, 083520 (2002), arXiv:astro-ph/0110460.",
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00c11ce62d9d0baba43f2eb4dd670d00ca85af6a | subsection | 3 | 11 | Body | The requirement for V is that it satisfies the
slow roll conditions |V^{\prime }/V|<1, |V^{\prime \prime }/V|<1, where a prime denotes
derivative w.r.t. to \phi , at the inflation epoch and at present
time for DE. We also take V such that \phi evolves through
regions where instant preheating is possible, e.g. V(\phi =0... | {
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e78a9dbb46f3175fa925502ff7da16f2e4d9c34f | subsection | 4 | 11 | Body | In the limit where the decaying particle is non-relativistic
with E_a\simeq m_a\gg m_b, p_b\simeq E_b then eq.(REF ) becomesOn the other hand
if all particles involved are relativistic and in TE then eq.(REF )
with n_a= c_nT^3, c_n=g_a\zeta (3)/\pi ^2 and E_a=T is\widetilde{c}_{ab}=c_{ab}c_n^{a-1}.
In quantum field the... | {
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} | 0807.1880 | Inflation-Dark Energy unified through Quantum Regeneration | [
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5b27e4bbe59179f44f5ec6d3b63faa0fc4ba948e | subsection | 5 | 11 | Body | If we take a polynomial potential V_{int}(\phi ,\varphi )=g\,\phi ^m\varphi ^n
with arbitrary values of m,n and use eq.(REF ) we have
M_{ab}=\frac{m!n!}{a!(m-a)!b!(n-b)!}\;g\phi ^{m-a}\varphi ^{n-b} and
eq.(REF ) becomes
\Gamma _{ab} =\Gamma _{12} \Gamma _i^{a-1}\Gamma _f^{b-2} with
\Gamma _0 \equiv c_0g^2\phi ^{2(m-1... | {
"cite_spans": []
} | 0807.1880 | Inflation-Dark Energy unified through Quantum Regeneration | [
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cb8cc3e9232431f40b4d440f858f6b8bc5056bf8 | subsection | 6 | 11 | Body | If the fields \chi ,\psi acquire a large mass
then \varphi will no longer be coupled at T< m_\chi since below
this temperature n_\chi , n_\psi are exponentially suppressed
and \Gamma /H will be smaller than one. However, the \varphi
temperature will still redshift as T\sim 1/a(t) since it is
relativistic.Let us now de... | {
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} | 0807.1880 | Inflation-Dark Energy unified through Quantum Regeneration | [
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be08d59a7bbad466789b3355c1ab0532e06ed3cf | subsection | 7 | 11 | Body | After inflation the energy
density \rho _\phi redshifts with an equation of state w_\phi \ne -1
and m_\varphi \phi \approx 0 for \phi \approx 0 giving
|\dot{m}_\varphi /m^2_{\varphi }|\gg 1 in eq.(REF ).Universe Reheating– The
reheating of the universe takes place via a process
\varphi +\varphi \leftrightarrow \chi +\... | {
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6841de8f709acd233450eb9f31c3ae8c015443eb | subsection | 8 | 11 | Body | The fine structure constant of these
interactions are \alpha _I\equiv h^2/4\pi =E_I/4\pi and
\alpha _{BD}\equiv g^2/4\pi =E^2_I/4\pi which for E_I=100\,TeV
gives \alpha _I=10^{-14}, \alpha _{BD}=10^{-27} to be compared with
\alpha _{em}=1/137, the electromagnetic fine structure constant.
The constraint on light particl... | {
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... | 0807.1880 | Inflation-Dark Energy unified through Quantum Regeneration | [
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d75154668773124d4bbed2f2fedd48f047747ec0 | subsection | 9 | 11 | Body | The inflation, reheating and
back decay scales, using eq.(REF ) with q=10, areThe scale E_I is very interesting
since it is on the upper limit of susy. This inflationary scale E_I is
low compared to the standard 10^{16} GeV but it is large enough to have a
reheating temperature to produce all SM particles and it is
wit... | {
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ea3483bc42c4ae7fcad14383049c426412dbc741 | subsection | 10 | 11 | Body | Once \phi is
regenerated it will grow and its potential will start dominating the
universe with \phi =O(1) for V\approx V_o, independent of its
initial conditions (tracker behavior). The slow roll conditions are
satisfied and the universe will enter an acceleration period or DE
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an... | {
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b8617c38682b39c86a84a811f37dcb484b27e9dc | abstract | 0 | 7 | Abstract | We demonstrate real-time detection of self-interfering electrons in a double
quantum dot embedded in an Aharonov-Bohm interferometer, with visibility
approaching unity. We use a quantum point contact as a charge detector to
perform time-resolved measurements of single-electron tunneling. With increased
bias voltage, th... | {
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} | 10.1021/nl801689t | 0807.1881 | Time-resolved detection of single-electron interference | [
"S. Gustavsson",
"R. Leturcq",
"M. Studer",
"T. Ihn",
"K. Ensslin",
"D. C. Driscoll",
"A. C. Gossard"
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c109362ef612fc269691ded75c5367241420e348 | subsection | 1 | 7 | Body | Time-resolved detection of single-electron interference
S. Gustavsson
simongus@phys.ethz.ch
R. Leturcq
M. Studer
T. Ihn
K. Ensslin
Solid State Physics Laboratory, ETH Zürich, CH-8093 Zürich,
Switzerland
D. C. Driscoll
A. C. Gossard
Materials Departement, University of California, Santa
Barbara, CA-93106, USAWe demonstr... | {
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a1a23b3156034a5e901b7b7d4e189af5f3cf72e0 | subsection | 2 | 7 | Body | Upon arriving in QD2, the electrons are detected in real-time by
operating a near-by quantum point contact (QPC) as a charge detector
. Coulomb blockade prohibits more than one excess
electron to populate the structure, implying that the first electron
must leave to the drain before a new one can enter. This enables
ti... | {
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a89b5fcba1b14d4248bcc5d2d582d4dee775a07e | subsection | 3 | 7 | Body | The solidlines are tunneling rates expected from sequential tunneling,while the dashed line is a fit to the cotunneling model of Eq. ().Parameters are given in the text. The data was taken with B=340~\mathrm {mT}.(c) Energy level configuration of the DQD at the point marked by II in (a, b). Electron transport from sour... | {
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c52bdcefc68ee4ed405e6c39129e5d46a4670298 | subsection | 4 | 7 | Body | We emphasize that
Eq. (REF ) is valid only if \delta _a, \delta _b \gg t_a, t_b
and if sequential transport is sufficiently suppressed, i.e. in the
range 46~\mathrm {mV} < V_\mathrm {G1} < 48.6~\mathrm {mV} of
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b24e6549e2da3606fbcced991580a216cec6f3cb | subsection | 5 | 7 | Body | REF (a),
i.e., to the energy of the states in QD1. The measurement shows a
general shift of the DQD energy with the applied B-field, which we
attribute to changes of the orbital wavefunctions in the individual
QDs. Within the cotunneling region, \Gamma _\mathrm {in} shows
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64890990a69b2223aec41f94ad6f836eda7ba765 | subsection | 6 | 7 | Body | At V_\mathrm {QPC} = 400~\mathrm {\mu V}, the current
through the QPC is approximately 10~\mathrm {nA}. This gives an
average time delay between two electrons passing the QPC of
e/I_\mathrm {QPC} \sim \! 16~\mathrm {ps}. Since this is ten times
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de07131bd3ea57c8c8baa684ad3ee120455c865f | abstract | 0 | 47 | Abstract | Granular elasticity, an elasticity theory useful for calculating static
stress distribution in granular media, is generalized to the dynamic case by
including the plastic contribution of the strain. A complete hydrodynamic
theory is derived based on the hypothesis that granular medium turns
transiently elastic when def... | {
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"Yimin Jiang",
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97a72897a766444d4fe095e458e9b7f5bbc040a7 | subsection | 1 | 47 | Introduction | Widespread interests in granular media were aroused among
physicists a decade ago, stimulated in large part by
review articles revealing the intriguing and improbable
fact that something as familiar as sand is still rather
poorly understood , , , . The
resultant collective efforts have since greatly enhanced
our unders... | {
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e3ef3b3c0fa00ccf7bc4d7c956dc519ea4b2d3cf | subsection | 2 | 47 | Introduction | Transiently elastic media such as polymers are under
active consideration at
present , , , .The main advantage of the hydrodynamic approach is its
stringency. In the Truesdell approach, apart from
objectivity, few general constraints exist for the
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bb2d781756109cd1a8faf58bb652ebe52ea42924 | subsection | 3 | 47 | Introduction | This is the reason granular media can sustain static
stress only when at rest, but looses it gradually when
being tapped or sheared. And our assumption is, this
happens similarly no matter how the grains jiggle and
slide, and we may therefore parameterize their stochastic
motion as a scalar T_g. Our guiding notion is
t... | {
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3b136c3f9c7521d11443e7b1492c7adb5a61ee11 | subsection | 4 | 47 | Introduction | They contain
innumerable internal degrees of freedom that are
neglected in mesoscopic
models , , , , . For
instance, phonons contained in individual grains do
explore the phase space and arrive at a distribution
appropriate for the ambient temperature. Jamming fixes
only a few out of many, many degrees of freedom.
Real... | {
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90bb30a205a2a519efb96c943050029ee949a318 | subsection | 5 | 47 | Introduction | Section presents the formal derivation of the
hydrodynamic theory. The resulting equations are then
applied to reproduce the hypoplastic model in
section . Finally, section gives a
brief summary. | {
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a15e8f62ee0ffc18ebdb175fa2a896d358b9dcbd | subsection | 6 | 47 | Sand – a Transiently Elastic Medium | Granular media possess different phases that, depending
on the grain's ratio of elastic to kinetic energy, may
loosely be referred to as gaseous, liquid and solid.
Moving fast and being free most of the time, the grains
in the gaseous phase have much kinetic, but next to none
elastic,
energy , , , , . In the
denser liq... | {
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43c25241098c1efa9fceca9c8308145cc0e39bef | subsection | 7 | 47 | Sand – a Transiently Elastic Medium | The elastic coefficient \mathcal {B}, a measure of
overall rigidity, is a function of the density \rho .
Assuming a uniform \rho (hence a spatially constant
\cal B), the stress at the bottom of a sand pile is (as
one would expect) maximal at the center. But a stress dip
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c92d065850ab27a7c52e5073c44ac030502796ae | subsection | 8 | 47 | Jamming and Granular Equilibria | Liquid and solid equilibria are first described, then
shown to correspond to the unjammed and jammed equilibria
of granular media. | {
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45a729901ff262bb5e6a1e9784a39e4153b35ed6 | subsection | 9 | 47 | Liquid Equilibrium | In liquid, the conserved energy density
w(s,\rho ,g_i) depends on the densities of entropy
s, mass \rho , and momentum g_i=\rho v_i. The
dependence on g_i is universal, given simply byw(s,\rho ,g_i)=w_0(s,\rho )+g_i^2/(2\rho ),leaving the rest-frame energy w_0 to contain the
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33d6bb4c7a98b1767bfbbeec5fb93ed8178fc04f | subsection | 10 | 47 | Liquid Equilibrium | We focus on
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with G_k=-\nabla _i\phi the gravitational constant
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dea28b92ad7b2b24f96eeef704a5164da513ca8b | subsection | 11 | 47 | Solid Equilibrium | In solids, if the subtle effect of mass defects is
neglected, density is not an independent variable and
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d7c2e64e90002ad9d8958fae560a201deeba1d36 | subsection | 12 | 47 | Granular Equilibria | Depending on whether T_g is zero or finite, sand
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532123bb5f47dd1aca1a216960c1c599744ae548 | subsection | 13 | 47 | Granular Temperature | Granular temperature is not a new concept. Haff, at the
same time Jenkins and
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4d8361a66c5f18c9743771e5b1d5169a260d8347 | subsection | 14 | 47 | The Equilibrium Condition for | The energy change {\rm d}w from all microscopic,
implicit variables is generally subsumed as T{\rm d}s,
with s the entropy and T\equiv \partial w_0/\partial s
its conjugate variable. From this, we divide out the
intergranular energy of the random motion of the grains,
denoting it as T_g{\rm d}s_g,{\rm d}w_0=T{\rm d}(s-... | {
"cite_spans": []
} | 10.1007/s10035-009-0137-3 | 0807.1883 | Granular Solid Hydrodynamics | [
"Yimin Jiang",
"Mario Liu"
] | [
"cond-mat.soft"
] | 2,008 | en | Physics | [
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54e0583acc0b85aeabb7022d45daf8f83edad1e2 | subsection | 15 | 47 | The Equation of Motion for | Being a macroscopic, non-hydrodynamic variable, s_g
must first of all obey a relaxation equation,
-{\partial _t}s_g =\gamma \partial f/\partial s_g=\gamma \bar{T}_g. Since this relaxation is typically slow, s_g
also displays characteristics of a quasi-conserved
quantity, and removal of local accumulations is accounted
... | {
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] | [
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18b301adb04109921d1f5f2fe18012eef4ef5010 | subsection | 16 | 47 | The Equation of Motion for | As discussed in
section REF , these are, in
addition, the vanishing of \pi _{ij},
\nabla _j\pi _{ij}, and \bar{T}_g, hence we haveR=\eta v_{ij}^0v_{ij}^0+\zeta v_{\ell \ell }^2+
\kappa (\nabla _iT)^2+\gamma \bar{T}_g^2
\\+\beta (\pi ^0_{ij})^2+\beta _1\pi _{\ell \ell }^2
+\beta ^P(\nabla _j\pi _{ij})^2.Three additional... | {
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298f5787bf2a69e03c6187744ce9cc4dff3a093c | subsection | 17 | 47 | The Equation of Motion for | A direct consequence for the
stationary case, R_g=0, is\gamma \bar{T}_g^2=\eta _g v_{ij}^0v_{ij}^0+\zeta _g
v_{\ell \ell }^2,quantifying how much \bar{T}_g\equiv T_g-T is excited by
shear or compressional flows.In dry sand, the granular viscosities \eta _g,\zeta _g
probably dominate, while \eta ,\zeta are insignificant... | {
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} | 10.1007/s10035-009-0137-3 | 0807.1883 | Granular Solid Hydrodynamics | [
"Yimin Jiang",
"Mario Liu"
] | [
"cond-mat.soft"
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d082a0e9d8e23050d36158f0d278218489f96236 | subsection | 18 | 47 | Two Fluctuation-Dissipation Theorems | There are many in the granular community who dispute the
validity of the Onsager reciprocity relation in granular
media, enlisting any of the following three reasons:
(1) The fluctuation-dissipation theorem (fdt) does
not hold. (2) The microscopic dynamics is not reversible.
(3) Sand is too far off equilibrium.Careful ... | {
"cite_spans": []
} | 10.1007/s10035-009-0137-3 | 0807.1883 | Granular Solid Hydrodynamics | [
"Yimin Jiang",
"Mario Liu"
] | [
"cond-mat.soft"
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b6a057d73efde4fe601d5e466dd5b03a4abf1015 | subsection | 19 | 47 | Elastic and Plastic Strain | As discussed in section , the elastic
strain u_{ij} accounts for the deformation of
individual grains, while their rolling and sliding is
described by the plastic strain p_{ij}. Together,
they form the total strain\varepsilon _{ij}= u_{ij}+p_{ij}.The elastic energy w(u_{ij}) is a function of
u_{ij}, not of \varepsilon ... | {
"cite_spans": []
} | 10.1007/s10035-009-0137-3 | 0807.1883 | Granular Solid Hydrodynamics | [
"Yimin Jiang",
"Mario Liu"
] | [
"cond-mat.soft"
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2efac6abd8d1186840cd2bf9060037db4ebc1ad0 | subsection | 20 | 47 | Elastic and Plastic Strain | Assuming (for simplicity) a stationary granular
temperature, or
T_g^2=(\eta _g/\gamma ){v_{ij}v_{ij}}\equiv (\eta _g/\gamma )||v_{s}||^2,
see Eq (REF ), we obtain from Eq (REF )
the equation,\partial _t
u_{ij}-v_{ij}\sim ||v_{s}||(-u_{ij})\sqrt{\eta _g/\gamma }\,,the rate-independent structure of which closely resemble... | {
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... | 10.1007/s10035-009-0137-3 | 0807.1883 | Granular Solid Hydrodynamics | [
"Yimin Jiang",
"Mario Liu"
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5a668e36645fb5c15f869f5cea5007ce1da771da | subsection | 21 | 47 | The Granular Free Energy | As explained in the Introduction, the structure of the
hydrodynamic theory is determined by general principles,
especially energy and momentum conservation, but the
explicit form of the energy w is not. Although w does
possess features that it must always satisfy, most of its
functional dependence reflects the specific... | {
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... | 10.1007/s10035-009-0137-3 | 0807.1883 | Granular Solid Hydrodynamics | [
"Yimin Jiang",
"Mario Liu"
] | [
"cond-mat.soft"
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387d571b25772a9a47cbbf35da840e577ed02302 | subsection | 22 | 47 | The Granular Free Energy | This neglects
effects such as thermal expansion that, however, may be
added when necessary.)Being cohesionless, the grains possess no interaction
energy, f_0(T,\rho ) is therefore the sum of the free
energy in each of the grains,f_0(T,\rho )=\langle F_1(T)/m\rangle \rho ,where F_1 is the free energy of a single grain, ... | {
"cite_spans": []
} | 10.1007/s10035-009-0137-3 | 0807.1883 | Granular Solid Hydrodynamics | [
"Yimin Jiang",
"Mario Liu"
] | [
"cond-mat.soft"
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1468bc085e9fbed3de605cdffe96764ea9c95998 | subsection | 23 | 47 | The Elastic Energy | The elastic part of the free energy, Eq (), has
previously been successfully tested under varying
circumstances, cf. the discussion in
section , below Eq (). It is not
analytic in the elastic strain, but does contain the
lowest order terms. As it takes some deliberation to
arrive at its density dependence and the terms... | {
"cite_spans": []
} | 10.1007/s10035-009-0137-3 | 0807.1883 | Granular Solid Hydrodynamics | [
"Yimin Jiang",
"Mario Liu"
] | [
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ad7925e0468dc75a369ab8250f9ab4e56a894e7b | subsection | 24 | 47 | Density Dependence of | We shall take {\cal B} as density dependent, but not
\xi : Since the Coulomb yield line is approximately
independent of the density, so must the coefficient \xi
be, see Eq (REF ). Granular sound velocity was
measured by Hardin and Richart , who found
it linear in the void ratio, c\sim 2.17-e. Given
Eq (), the velocity... | {
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"Yimin Jiang",
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cd675daba870ca486bbbc5678d42372e2be6b742 | subsection | 25 | 47 | Density Dependence of | Theplots are calculated with \rho _{\ell c}^*=0.445\rho _G, \rho _{pc}=0.645\rho _G (implying \rho _{\ell p}=0.555\rho _G), and {\cal B}_0=7000 Mpa, appropriatefor Ham River sand .]Alas, these points are more easily stated than
combined in an energy expression, and no continuous
\cal B seems feasible: If analytic, \cal... | {
"cite_spans": []
} | 10.1007/s10035-009-0137-3 | 0807.1883 | Granular Solid Hydrodynamics | [
"Yimin Jiang",
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3cf78c9b15395b1fb87e99056af2f7194e945d7d | subsection | 26 | 47 | Higher-Order Strain Terms | Next, we consider the unjamming transition in connection
with compaction by pressure increase, the fact that
denser sand can sustain more compression before getting
unjammed, before elastic solutions become unstable: See
the dotted line of Fig REF -(a), depicting a
well-known empirical formula from soil
mechanics , , ,... | {
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ab4aab95896949904c453fabfec22f67fee901c8 | subsection | 27 | 47 | Higher-Order Strain Terms | Given this lack of reliable
data, we decided against the expansion, Eq (REF ),
and opted for a flexible “cap function," \cal C of
Eq (REF ), capable of accounting for any possible
cap-like unjamming transitions,2{\cal C}=1+\tanh [(\Delta _0-\Delta )/\Delta _1],
\quad \text{where}\quad \\ \Delta _0=k_1\rho -k_2u_s^2-k_3... | {
"cite_spans": []
} | 10.1007/s10035-009-0137-3 | 0807.1883 | Granular Solid Hydrodynamics | [
"Yimin Jiang",
"Mario Liu"
] | [
"cond-mat.soft"
] | 2,008 | en | Physics | [
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9b4db5e8aed92ba049b4ca74c68c3e7e16a863f2 | subsection | 28 | 47 | Higher-Order Strain Terms | As linear transformations do not alter the
convexity property of any function, we may take the
energy as w_7(\rho , \Delta , x_{1-5}) where
x_1\equiv \sqrt{2 }u_{xy}, x_2\equiv \sqrt{2}u_{xz},
x_3\equiv \sqrt{2}u_{yz},
x_4\equiv (u_{xx}-u_{zz})/\sqrt{2},
x_5\equiv (u_{xx}-2u_{yy}+u_{zz})/\sqrt{6}. The
characteristic po... | {
"cite_spans": []
} | 10.1007/s10035-009-0137-3 | 0807.1883 | Granular Solid Hydrodynamics | [
"Yimin Jiang",
"Mario Liu"
] | [
"cond-mat.soft"
] | 2,008 | en | Physics | [
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9bbb50a28ac87066b8a4321b0636967014a1a029 | subsection | 29 | 47 | Pressure Contribution From Agitated Grains | Agitated grains are known to exert a pressure in granular
liquid. Using the model of ideal gas (better:
non-interacting atoms with excluded volumes), with
w_2\sim {\rho T_g} denoting the energy density of
agitated grains, the pressure expression,P_T(\rho , T_g)\sim {w_2}/({1-\rho /\rho _{cp}}),was employed and found to... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 452,
"openalex_id": "",
"raw": "L. Bocquet, J. Errami, and T. C. Lubensky, Hydrodynamic Model for a Dynamical Jammed-to-Flowing Transition in Gravity Driven Granular Media, Phys. Rev. Lett., 89, 184301 (2002).",
"source_ref_... | 10.1007/s10035-009-0137-3 | 0807.1883 | Granular Solid Hydrodynamics | [
"Yimin Jiang",
"Mario Liu"
] | [
"cond-mat.soft"
] | 2,008 | en | Physics | [
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df553d37060190ab94921a27803e2c306bee3ba8 | subsection | 30 | 47 | Pressure Contribution From Agitated Grains | For instance, the yield condition
of Eq (REF ), with \xi =5/3, now reads\frac{\pi _s}{P_\Delta }=
\frac{\pi _s}{P-P_T}\le \sqrt{\frac{6}{5}},implying a smaller maximal \pi _s for given P. On the
other hand, the maximal value for the void ratio e is
larger when P_T is present: Any given e has a maximal
elastic compressi... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1504,
"openalex_id": "",
"raw": "P.A. Johnson, X. Jia, Nonlinear dynamics, granular media and dynamic earthquake triggering Nature, 437/6, 871 (2005).",
"source_ref_id": "adffca4dec81059527bc28e126d2036b2c1cf502",
"sta... | 10.1007/s10035-009-0137-3 | 0807.1883 | Granular Solid Hydrodynamics | [
"Yimin Jiang",
"Mario Liu"
] | [
"cond-mat.soft"
] | 2,008 | en | Physics | [
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d7cdaa41c70ad29a190c18b3a44dd37275c383bc | subsection | 31 | 47 | The Edwards Entropy | It is useful, with the free energy obtained in this
chapter in mind, to revisit the starting points of
Granular Statistical Mechanics (gsm), especially
the Edwards entropy . Taking the entropy
S(W,V) as a function of the energy W and volume V,
or {\rm d}S=(1/T){\rm d}W+(P/T){\rm d}V, it argues that
a mechanically stabl... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 173,
"openalex_id": "",
"raw": "S.F. Edwards, R.B.S. Oakeshott, Theory of powders, Physica A 157, 1080 (1989); S.F. Edwards, D.V. Grinev, Statistical Mechanics of Granular Materials: Stress Propagation and Distribution of Contact ... | 10.1007/s10035-009-0137-3 | 0807.1883 | Granular Solid Hydrodynamics | [
"Yimin Jiang",
"Mario Liu"
] | [
"cond-mat.soft"
] | 2,008 | en | Physics | [
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b69c1f17d3eb154175b401876951ed855f1fc492 | subsection | 32 | 47 | Derivation | We take the conserved energy w(s,s_g,\rho ,g_i,u_{ij})
of granular media to depend on entropy s, granular
entropy s_g, density \rho , momentum density g_i,
and the elastic strain u_{ij}. Defining the conjugate
variables as T\equiv \partial w/\partial s, \bar{T}_g\equiv T_g-T\equiv \partial w/\partial s_g [see
Eq (REF )... | {
"cite_spans": []
} | 10.1007/s10035-009-0137-3 | 0807.1883 | Granular Solid Hydrodynamics | [
"Yimin Jiang",
"Mario Liu"
] | [
"cond-mat.soft"
] | 2,008 | en | Physics | [
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45c7065460c7f120249c5909adc690f53e15d69a | subsection | 33 | 47 | Derivation | Next, we introduce
\sigma ^D_{ij}+\Sigma ^D_{ij}, as\sigma _{ij}\equiv (-\tilde{f}+\mu \rho )
\delta _{ij}-(\sigma ^D_{ij}+\Sigma ^D_{ij})\\
+\pi _{ij}-\pi _{ik}u_{jk}-\pi _{jk}u_{ik},where \tilde{f}\equiv w_0-Ts-\bar{T}_gs_g, as in
Eq (REF ,REF ). This is simply a definition of
\sigma ^D_{ij}+\Sigma ^D_{ij}, which tra... | {
"cite_spans": []
} | 10.1007/s10035-009-0137-3 | 0807.1883 | Granular Solid Hydrodynamics | [
"Yimin Jiang",
"Mario Liu"
] | [
"cond-mat.soft"
] | 2,008 | en | Physics | [
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28efbb6fd3fea398120367b2064ed646ee3e7356 | subsection | 34 | 47 | Derivation | We take the first
line to yield the energy flux, Q_i, and the next two
lines to vanish independently,Q_i&=&Tf_i+\bar{T}_gF_i+\mu \rho v_i
+v_j\sigma _{ij}-y_j\pi _{ij}, \\
R&=&f_i^D\nabla _iT+\sigma _{ij}^Dv_{ij}
+y_i\nabla _j\pi _{ij} +X_{ij}\pi _{ij}+\gamma \bar{T}_g^2,\\
R_g&=&\Sigma _{ij}^Dv_{ij} +F_i^D\nabla _i\ba... | {
"cite_spans": []
} | 10.1007/s10035-009-0137-3 | 0807.1883 | Granular Solid Hydrodynamics | [
"Yimin Jiang",
"Mario Liu"
] | [
"cond-mat.soft"
] | 2,008 | en | Physics | [
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3f6a2407ef4a76445955d3d9752adbceae4ad8e1 | subsection | 35 | 47 | Derivation | Given f_i^D,
F_i^D, \sigma _{ij}^D, \Sigma _{ij}^D, y_i, X_{ij}, the
structure of all currents in the set of equation,
Eqs (REF ,,,),
are known. The question that remains is whether these
expressions are unique. For simpler hydrodynamic
theories, such as for isotropic liquid, nematic liquid
crystal, or elastic solid, t... | {
"cite_spans": []
} | 10.1007/s10035-009-0137-3 | 0807.1883 | Granular Solid Hydrodynamics | [
"Yimin Jiang",
"Mario Liu"
] | [
"cond-mat.soft"
] | 2,008 | en | Physics | [
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a2d778aa9e76bd4e4f59c47b93ba01ef84f0a931 | subsection | 36 | 47 | Results | Collecting the terms derived above, in section REF ,
the equations of gsh, with \sigma _{ij} valid to
lowest order in strain, are\partial _t \rho +\nabla _i(\rho v_i)=0,\qquad \qquad \quad
\\{\rm d}_t
u_{ij}=(1-\alpha )v_{ij}-{u _{ij}^0}/\tau -{u_{\ell \ell }\,\delta _{ij}}/{\tau _1}\qquad \\ -(u_{ik}\nabla _jv_k+
\na... | {
"cite_spans": []
} | 10.1007/s10035-009-0137-3 | 0807.1883 | Granular Solid Hydrodynamics | [
"Yimin Jiang",
"Mario Liu"
] | [
"cond-mat.soft"
] | 2,008 | en | Physics | [
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587f700fa52f0333024443e6b02d20663db85791 | subsection | 37 | 47 | Results | This is not true for
granular media, which typically possess more involved
functional dependence – especially concerning the \bar{T}_g\rightarrow 0 limit, which does not have a counter part in
other systems. This is one of the less recognized
reasons, we believe, underlying the complexity of
granular systems.In section... | {
"cite_spans": []
} | 10.1007/s10035-009-0137-3 | 0807.1883 | Granular Solid Hydrodynamics | [
"Yimin Jiang",
"Mario Liu"
] | [
"cond-mat.soft"
] | 2,008 | en | Physics | [
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