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[EQUATION] As [MATH] , this solution tends to the [MATH] single-mode state ( 2.11 ). Thanks to the transformation 2.1 ), one can set [MATH] , then [MATH] yields the normalized state ( 1.3 ). |
As is explained above, the geometric sequence ( 2.16 ) is a maximizer of [MATH] for fixed [MATH] therefore, we call it the ground state Nonlinear stability of the ground state has been proven in |
where the degeneracy due to the parameter [MATH] has been controlled with the use of conserved quantities [MATH] and [MATH] 2.2.2. Twisted state |
If [MATH] but [MATH] , equation ( 2.13 ) is satisfied with [MATH] . Then, equation ( 2.14 ) yields [MATH] whereas equation ( 2.15 ) is identically satisfied with [MATH] and [MATH] Parameterizing [MATH] and bringing all together yield the twisted state |
[EQUATION] As [MATH] , this solution tends to the [MATH] single-mode state ( 2.11 ). Thanks to transformation ( 2.1 ), one can set [MATH] , then [MATH] |
2.2.3. Pair of stationary states If [MATH] and [MATH] , then [MATH] can be eliminated from system 2.14 ) and ( 2.15 ), which results in the algebraic equation |
[EQUATION] The first factor corresponds to the twisted state ( 2.17 ). Computing [MATH] from the quadratic equation in the second factor yield two roots: |
[EQUATION] The real roots exist for [MATH] which is true for [MATH] or equivalently, for [MATH] . The pair of solutions can be parameterized as follows: |
[EQUATION] where [MATH] is arbitrary. Using equations ( 2.13 ) and ( 2.14 ) again, we get [EQUATION] As [MATH] , the branch with the upper sign converges to the [MATH] single-mode state |
[EQUATION] while the branch with the lower sign converges to the [MATH] single-mode state [EQUATION] where [MATH] has been rescaled as [MATH] The solutions ( 2.12 ), ( 2.20 ), and ( 2.21 have been cast as the stationary states ( 1.4 ) and ( 1.5 ). By using our bifurcation analysis, we will prove that these stationary s... |
We summarize that the four explicit families ( 2.16 ), ( 2.17 ), and ( 2.21 form a complete set of stationary states given by ( 2.12 ). |
2.3. Other stationary states Other stationary states were constructed in from the generating function [MATH] given by the power series expansion: |
[EQUATION] The conformal flow ( 1.1 ) can be rewritten for [MATH] as the integro-differential equation [EQUATION] The stationary solutions expressed by ( 2.9 ) yield the generating function in the form |
[EQUATION] where [MATH] is a function of one variable given by [MATH] The family of stationary solutions expressed by ( 2.12 ) is |
[EQUATION] Additionally, any finite Blaschke product [EQUATION] yields a stationary state with [MATH] and [MATH] for [MATH] . If [MATH] and [MATH] the generating function ( 2.28 ) yields a stationary solution of the system ( 2.10 ) in the form |
[EQUATION] This solution can be viewed as a continuation of the [MATH] single-mode state ( 2.11 ) in [MATH] Another family of stationary states is |
[EQUATION] where [MATH] [MATH] , and [MATH] is arbitrary. When [MATH] , the function ( 2.30 ) generates the geometric sequence 2.16 ). When [MATH] , the function ( 2.30 ) generates another stationary solution |
[EQUATION] which is a continuation of the [MATH] single-mode state ( 2.11 ) in [MATH] Thanks to transformation ( 2.1 ), one can set [MATH] , then [MATH] The new solutions ( 2.29 ) and ( 2.31 ) appear in the local bifurcation analysis from the [MATH] single-mode state ( 2.11 ). |
3. Bifurcations from the lowest eigenmode We restrict our attention to the real-valued solutions of the stationary equations ( 2.10 ), which satisfy the system of algebraic equations |
[EQUATION] Here we study bifurcations of stationary states from the lowest eigenmode given by ( 2.11 ) with [MATH] Without loss of generality, the scaling transformation ( 2.1 ) yields [MATH] and [MATH] By setting [MATH] with real-valued perturbation [MATH] , we rewrite the system ( 3.1 ) with [MATH] in the perturbativ... |
[EQUATION] where [MATH] is a diagonal operator with entries [EQUATION] and [MATH] includes quadratic and cubic nonlinear terms: [EQUATION] |
We have the following result on the nonlinear terms. Lemma 1 Fix an integer [MATH] . If [EQUATION] then [EQUATION] Proof. Let us inspect [MATH] in ( 3.4 ) for [MATH] with [MATH] and |
[MATH] . Since [MATH] only if [MATH] is multiple of [MATH] , then [MATH] in the first term of ( 3.4 ) for all [MATH] which are not a multiple of [MATH] Similarly, since [MATH] only if [MATH] is multiple of [MATH] , then [MATH] |
in the second term of ( 3.4 ). Finally, since [MATH] and [MATH] only if [MATH] and [MATH] are multiple of [MATH] , then [MATH] in the third term of ( 3.4 ). Hence, all terms of ( 3.4 ) are identically zero for [MATH] with [MATH] and |
[MATH] Bifurcations from the lowest eigenmode are identified by zero eigenvalues of the diagonal operator [MATH] Lemma 2 There exists a sequence of bifurcations at [MATH] |
with [EQUATION] where all bifurcation points are simple except for [MATH] Proof. The sequence of values ( 3.7 ) yield zero diagonal entries in ( 3.3 ). To study if the bifurcation points are simple, we consider solutions of |
[MATH] for [MATH] . This equation is equivalent to [MATH] , which has only two solutions [MATH] Therefore, all bifurcation points are simple except for the double point |
[MATH] The standard Crandall–Rabinowitz theory can be applied to study bifurcation from the simple zero eigenvalue. Theorem 1 Fix [MATH] or an integer [MATH] . There exists a unique branch of solutions [MATH] |
to system ( 3.1 ) with [MATH] , which can be parameterized by small [MATH] such that [MATH] is smooth in [MATH] and [EQUATION] Proof. |
Let us consider the decomposition: [EQUATION] where [MATH] is defined by ( 3.7 ), [MATH] is arbitrary and [MATH] is set from the orthogonality condition [MATH] Let [MATH] . The system ( 3.2 ) is decomposed into the invertible part |
[EQUATION] and the bifurcation equation [EQUATION] Since [EQUATION] we have [MATH] for every [MATH] and [MATH] as [MATH] Therefore, the Implicit Function Theorem can be applied to |
[EQUATION] where as is defined by ( 3.10 ), [MATH] is smooth in its variables, [MATH] for every [MATH] [MATH] , and [MATH] For every small [MATH] and small [MATH] , there exists a unique small solution of system ( 3.10 ) in [MATH] such that |
[MATH] . Denote this solution by [MATH] Since [EQUATION] one power of [MATH] is canceled in ( 3.11 ) and the Implicit Function Theorem can be applied to |
[EQUATION] where as is defined by ( 3.11 ), [MATH] is smooth in its variables, [MATH] , and [EQUATION] For every small [MATH] , there is a unique small root [MATH] |
of the bifurcation equation ( 3.11 ) such that [MATH] thanks to ( 3.13 ). Remark 1 The unique branch bifurcating from [MATH] coincides with the normalized ground state ( 1.3 ) and yields the exact result |
[MATH] . The small parameter [MATH] is defined in terms of the small parameter [MATH] by [MATH] Remark 2 For the unique branch bifurcating from [MATH] with [MATH] we claim that [MATH] [MATH] if and only if [MATH] is a multiple of [MATH] Indeed, by Lemma , the system ( 3.2 ) can be reduced for a new sequence [MATH] wher... |
[MATH] is a simple bifurcation point of this reduced system. By Theorem 3.8 there exists a unique branch of solutions of this reduced system with the bounds ( 3.8 ). Therefore, by uniqueness, the bifurcating solution satisfies the reduction of Lemma , that is, |
[MATH] [MATH] if and only if [MATH] is a multiple of [MATH] Remark 3 There are at least three branches bifurcating from the double point [MATH] . One branch satisfies 3.5 ) with [MATH] , the other branch satisfies ( 3.5 ) with [MATH] and the third branch is given by the explicit solution 1.4 ) with ( 1.5 ) for the uppe... |
As is well-known , normal form equations have to be computed in order to study branches bifurcating from the double zero eigenvalue at the bifurcation point [MATH] |
Theorem 2 Fix [MATH] There exist exactly three branches of solutions [MATH] to system ( 3.1 ) with [MATH] , which can be parameterized by small [MATH] such that [MATH] is smooth in [MATH] and |
[EQUATION] The three branches are characterized by the following: (i) [MATH] [MATH] (ii) [MATH] [MATH] (iii) [MATH] [MATH] and the branch (iii) is double degenerate up to the reflection [MATH] |
Proof. Let us consider the decomposition: [EQUATION] where [MATH] [MATH] are arbitrary, and [MATH] are set from the orthogonality condition [MATH] The system ( 3.2 ) is decomposed into the invertible part |
[EQUATION] and the bifurcation equations [EQUATION] By the Implicit Function Theorem (see the proof of Theorem 3.8 ), there exists a unique map |
[MATH] for small [MATH] such that equations ( 3.16 ) are satisfied and [MATH] . Denote this solution by [MATH] Compared to the proof of Theorem 3.8 , it is now more difficult to consider solutions of the system of two algebraic equations ( 3.17 ). In order to resolve the degeneracy of these equations, we have to comput... |
[MATH] up to the cubic terms in [MATH] under the apriori assumption [MATH] Substituting the expansion for [MATH] into the system ( 3.17 and expanding it up to the quartic terms in [MATH] , we will be able to confirm the apriori assumption |
[MATH] and to obtain all solutions of the system ( 3.17 for [MATH] near [MATH] First, we note that if [MATH] , then [MATH] for every [MATH] , where |
[MATH] denotes terms of the cubic and higher order in [MATH] Thanks to the cubic smallness of [MATH] for [MATH] , we can compute [MATH] for [MATH] up to and including the cubic order: |
[EQUATION] By using the system ( 3.16 ) and the quadratic approximations for [MATH] we obtain [MATH] up to and including the cubic order: |
[EQUATION] Next, we compute [MATH] for [MATH] up to and including the quartic order: [EQUATION] By using the quadratic and cubic approximations for [MATH] , we rewrite the system ( 3.17 ) up to and including the quartic order: |
[EQUATION] By Lemma , if [MATH] , then [MATH] whereas if [MATH] , then [MATH] Hence, the system ( 3.23 ) can be rewritten in the equivalent form: |
[EQUATION] We are looking for solutions to the system ( 3.26 ) with [MATH] , because [MATH] follows from the system ( 3.16 ). There exist three nontrivial solutions to the system ( 3.23 ): |
(I) [MATH] [MATH] , and [MATH] (II) [MATH] [MATH] , and [MATH] (III) [MATH] [MATH] , and [EQUATION] All the solutions satisfy the apriori assumption [MATH] In order to consider persistence of these solutions in the system ( 3.26 ), we compute the Jacobian matrix: |
[EQUATION] We proceed in each case as follows: (I) The Jacobian is invertible for small [MATH] admitting the expansion in (I). By the Implicit Function Theorem, there exists a unique continuation of this root in the system ( 3.23 ). By Lemma with [MATH] , the bifurcating solution corresponds to the reduction with [MATH... |
(II) The Jacobian is invertible for small [MATH] admitting the expansion in (II). By the Implicit Function Theorem, there exists a unique continuation of this root in the system ( 3.23 ). By Lemma with [MATH] , the bifurcating solution corresponds to the reduction with [MATH] for every [MATH] hence [MATH] persists beyo... |
(III) Eliminating [MATH] from the system ( 3.29 ) yields the root finding problem: [EQUATION] which has only two solutions for [MATH] near [MATH] from the two roots of the quadratic equation [MATH] Computing |
[EQUATION] verifies that [MATH] is invertible at each of the two roots. By the Implicit Function Theorem, there exists a unique continuation of each of the two roots in the system ( 3.23 ). For the root with [MATH] , this yields the solution (iii). Thanks to the symmetry ( 2.3 ) with [MATH] , every solution with [MATH] |
can be uniquely reflected to the solution with [MATH] by the transformation [MATH] and [MATH] for [MATH] Hence the two solutions with [MATH] and [MATH] are the same branch (iii) up to reflection [MATH] |
No other branches bifurcate from the point [MATH] Remark 4 The branch (iii) bifurcating from [MATH] coincides with the exact solution ( 2.20 ) and ( 2.21 ) for the upper sign. Indeed, by normalizing [MATH] and taking the limit [MATH] as in ( 2.22 ), we obtain [MATH] [MATH] , and |
[MATH] . The parameter [MATH] and [MATH] are related to each other by means of the small parameter [MATH] with the definitions [MATH] and |
[MATH] , hence [MATH] Remark 5 Branches (i) and (ii) bifurcating from [MATH] can be obtained by the Crandall–Rabinowitz theory by reducing sequences on the constrained subspace of [MATH] by the constraints ( 3.5 with [MATH] and [MATH] respectively. The zero eigenvalue is simple on the constrained subspace of [MATH] , w... |
in a similar context. 4. Variational characterization of the bifurcating states Stationary states ( 2.9 ) with parameters [MATH] and [MATH] are critical points of the functional |
[EQUATION] where [MATH] [MATH] , and [MATH] are given by ( 1.2 ), ( 2.4 ), and ( 2.5 ). Let [MATH] , where [MATH] is a real root of the algebraic system ( 2.10 ), whereas [MATH] |
and [MATH] are real and imaginary parts of the perturbation. Because the stationary solution [MATH] is a critical point of [MATH] , the first variation of [MATH] |
vanishes at [MATH] and the second variation of [MATH] at [MATH] can be written as a quadratic form associated with the Hessian operator. In variables above, we obtain the quadratic form in the diagonalized form: |
[EQUATION] where [MATH] is the inner product in [MATH] and [MATH] is the induced norm. After straightforward computations, we obtain the explicit form for the self-adjoint operators |
[MATH] , where [MATH] is the maximal domain and [MATH] are unbounded operators given by [EQUATION] The following lemma gives variational characterization of the lowest eigenmode at the bifurcation points [MATH] in Lemma |
Lemma 3 The following is true: For [MATH] , the [MATH] single-mode state ( 2.11 ) is a degenerate saddle point of [MATH] with one positive eigenvalue, zero eigenvalue of multiplicity three, and infinitely many negative eigenvalues bounded away from zero. |
For [MATH] , the [MATH] single-mode state ( 2.11 ) is a degenerate minimizer of [MATH] with zero eigenvalue of multiplicity five and infinitely many positive eigenvalues bounded away from zero. |
For [MATH] with [MATH] , the [MATH] single-mode state ( 2.11 ) is a degenerate saddle point of [MATH] with [MATH] negative eigenvalues, zero eigenvalue of multiplicity three, and infinitely many positive eigenvalues bounded away from zero. |
Proof. By the scaling transformation ( 2.1 ), we take [MATH] and [MATH] for which the explicit form ( 4.3 ) yields [EQUATION] We note that [MATH] given by ( 3.3 ) and [MATH] is only different from [MATH] at the first diagonal entry at [MATH] |
(which is [MATH] instead of [MATH] ). Because [MATH] in ( 4.4 ) are diagonal, the assertion of the lemma is proven from explicit computations: |
If [MATH] , then [MATH] and [MATH] , which yields the result. If [MATH] , then [EQUATION] which yields the result. If [MATH] with [MATH] , then |
[EQUATION] For [MATH] and every [MATH] [MATH] For [MATH] [MATH] For [MATH] [MATH] . This yields the result. Remark 6 It is shown in Lemma 6.2 of |
that the normalized ground state ( 1.3 ) bifurcating from [MATH] has the same variational characterization as the [MATH] single-mode state, hence it is a triple-degenerate constrained maximizer of [MATH] subject to fixed [MATH] The triple degeneracy of the family ( 2.16 ) is due to two gauge symmetries ( 2.2 ) and 2.3 ... |
, the latter degeneracy is due to the additional symmetry ( 2.7 ). Indeed, by applying [MATH] with [MATH] to [MATH] and using the general transformation law derived in |
, we obtain another solution in the form: [EQUATION] which coincides with the ground state ( 1.3 ) after the definition [MATH] Remark 7 |
All branches bifurcating from [MATH] for [MATH] cannot be obtained by applying [MATH] with [MATH] to the [MATH] single-mode state [MATH] , because the latter state is independent of the values of [MATH] |
Remark 8 All branches bifurcating from [MATH] for [MATH] have too many negative and positive eigenvalues and therefore, they represent saddle points of [MATH] subject to fixed [MATH] and [MATH] |
It remains to study the three branches bifurcating from [MATH] . By Lemma there is a chance that some of these three branches represent constrained minimizers of [MATH] subject to fixed [MATH] and [MATH] One needs to consider how the zero eigenvalue of multiplicity five splits when [MATH] with [MATH] sufficiently small... |
at the three branches bifurcating from [MATH] Lemma 4 Consider the three bifurcating branches in Theorem for [MATH] . For every [MATH] with [MATH] sufficiently small, the following is true: |
(i) [MATH] [MATH] (ii) [MATH] [MATH] (iii) [MATH] [MATH] For each branch, [MATH] has a double zero eigenvalue, [MATH] has no zero eigenvalue, and the rest of the spectrum of [MATH] and [MATH] is strictly positive and is bounded away from zero. |
Proof. Substituting [MATH] and [MATH] into ( 4.3 ) yields [EQUATION] where [MATH] with [MATH] follows from the decomposition ( 3.15 ). The correction terms [MATH] are uniquely defined by parameters [MATH] . In what follows, we consider the three branches in Theorem separately. |
Case (i): [MATH] [MATH] Here we have [MATH] [MATH] [MATH] [MATH] , and [MATH] for [MATH] Substituting these expansions in ( 4.5 ), we obtain |
[EQUATION] where [EQUATION] Let us represent the first [MATH] -by- [MATH] matrix block of the operator [MATH] and truncate it by up to and including [MATH] terms. The corresponding matrix blocks denoted by |
[MATH] and [MATH] are given respectively by [EQUATION] and [EQUATION] Zero eigenvalue of [MATH] is double and is associated with the subspace spanned by [MATH] There are two invariant subspaces of [MATH] , one is spanned by [MATH] |
and the other one is spanned by [MATH] . This makes perturbative analysis easier. For the subspace spanned by [MATH] , the eigenvalue problem is given by |
[EQUATION] The small eigenvalue for small [MATH] is obtained by normalization [MATH] . Then, we obtain [EQUATION] and [EQUATION] |
For the subspace spanned by [MATH] , the eigenvalue problem is given by [EQUATION] The small eigenvalue for small [MATH] is obtained by normalization [MATH] . Then, we obtain |
[EQUATION] and [EQUATION] By the perturbation theory, for every [MATH] sufficiently small, [MATH] has two simple (small) negative eigenvalues. Other eigenvalues are bounded away from zero for small [MATH] and by Lemma all other eigenvalues of [MATH] are strictly positive. Hence, [MATH] |
Zero eigenvalue of [MATH] is triple and is associated with the subspace spanned by [MATH] For every [MATH] , a double zero eigenvalue of [MATH] exists due to the two symmetries ( 2.2 ) and ( 2.3 ). The two eigenvectors for the double zero eigenvalue of [MATH] are spanned by [MATH] There are two invariant subspaces of [... |
and the other one is spanned by [MATH] Since we only need to compute a shift of the zero eigenvalue of [MATH] , we only consider the subspace of [MATH] spanned by [MATH] For this subspace, the eigenvalue problem for [MATH] is given by |
By the perturbation theory, for every [MATH] sufficiently small, [MATH] has one simple (small) negative eigenvalue and the double zero eigenvalue. Other eigenvalues are bounded away from zero for small [MATH] and by Lemma all other eigenvalues of [MATH] are strictly positive. Hence, [MATH] |
Case (ii): [MATH] [MATH] Here we have [MATH] [MATH] [MATH] [MATH] [MATH] , and [MATH] for [MATH] Substituting these expansions in ( 4.5 ), we obtain |
and the other one is spanned by [MATH] For the subspace spanned by [MATH] , the eigenvalue problem is given by [EQUATION] The small eigenvalue for small [MATH] is obtained by normalization [MATH] . Then, we obtain |
[EQUATION] and [EQUATION] For the subspace spanned by [MATH] , the eigenvalue problem is given by [EQUATION] The small eigenvalue for small [MATH] is obtained by normalization [MATH] . Then, we obtain |
[EQUATION] and [EQUATION] By the perturbation theory, for every [MATH] sufficiently small, [MATH] has one simple small negative eigenvalue and one simple small positive eigenvalue. Other eigenvalues are bounded away from zero for small [MATH] and by Lemma all other eigenvalues of [MATH] are strictly positive. Hence, [M... |
Case (iii): [MATH] [MATH] Here we introduce [MATH] and write [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] , and [MATH] for [MATH] Substituting these expansions in ( 4.5 ), we obtain |
[MATH] and [MATH] are given respectively by [EQUATION] [EQUATION] and [EQUATION] [EQUATION] Zero eigenvalue of [MATH] is double and is associated with the subspace spanned by [MATH] Since no invariant subspaces of [MATH] exist, we have to proceed with full perturbative expansions. As a first step, we express [MATH] for... |
in terms of [MATH] for the subspace spanned by [MATH] [MATH] , and [MATH] . We assume that [MATH] for the small eigenvalues and neglect terms of the order [MATH] and higher. This expansion is given by |
[EQUATION] Next, we substitute these expansions to the third and fourth equations of the eigenvalue problem for [MATH] and again neglect terms of the order [MATH] and higher: |
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