Bifurcations of Nonconstant Solutions of the Ginzburg-Landau Equation

Hirano, Norimichi; Rybicki, Sławomir

doi:https://doi.org/10.1155/2012/560975

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2012 | Article ID 560975 | https://doi.org/10.1155/2012/560975

Bifurcations of Nonconstant Solutions of the Ginzburg-Landau Equation

Norimichi Hirano¹and Sławomir Rybicki²

Academic Editor: Victor M. Perez Garcia

Received24 Feb 2012

Revised25 Apr 2012

Accepted09 May 2012

Published06 Sept 2012

Abstract

We study local and global bifurcations of nonconstant solutions of the Ginzburg-Landau equation from the families of constant ones. As the topological tools we use the equivariant Conley index and the degree for equivariant gradient maps.

1. Introduction

Let us consider the following Ginzburg-Landau problem: where is an open, bounded domain with smooth boundary , is an outward normal to at and .

There is a vast literature on this problem. Bifurcations of solutions of the Ginzburg-Landau-type problems have been considered by many authors, see, for instance, [1–10] and references therein. Usually the authors study local bifurcations of nonzero solutions of problem (1.1) with Dirichlet boundary condition by using the Crandall-Rabinowitz bifurcation theorem, the Krasnosiel'ski bifurcation theorem for potential operators, the Lyapunov-Schmidt reduction, the center manifold theorem, the attractor bifurcation theorem, or the implicit function theorem. On the other hand, the global bifurcations of solutions of the one-dimensional Ginzburg-Landau model have been studied in [3]. Using the Brouwer degree the authors have proved the existence of a closed connected set of asymmetric solutions which connect the global curve of symmetric solutions to an asymmetric normal state solution.

Our goal is to study the existence of nonconstant solutions of problem (1.1) with Neumann boundary condition. We apply the equivariant bifurcation theory technique. First of all, we study families of constant solutions of problem (1.1) and describe them assuming that the norm of magnetic field is constant, that is, for all , where is the usual Euclidean norm. We distinguish two cases and . Next we find necessary and sufficient conditions for the existence of local and global bifurcation points of nonconstant solutions from these families. Problem (1.1) is -symmetric, that is, if is a solution of this problem, then is. Therefore, we consider solutions of (1.1) as critical orbits of -invariant functionals. The basic idea is to apply the -equivariant Conley index, see [11, 12], and the degree for -equivariant gradient maps, see [13–18], to obtain a local and global bifurcation of critical -orbits of these functionals. The choice of these invariants seems to be the best adapted to our theory. Since the Leray-Schauder degree is not applicable (see the remarks under Corollary 4.2) we have chosen the invariants which are suitable for the study of critical orbits of invariant functionals.

After this introduction our paper is organized as follows.

In Section 2 we have summarized without proofs the relevant abstract material on the equivariant bifurcation theory. In the next sections we have applied these abstract results to the study of local and global bifurcation of nonconstant solutions of problem (1.1). Since the Ginzburg-Landau equation is -symmetric, we consider in this section only -symmetric variational bifurcation problems. The notion of a local and global bifurcation of critical -orbits of families of -invariant -functionals has been introduced in Definition 2.1. The necessary condition for the existence of bifurcation points of critical -orbits has been formulated in Lemma 2.3. The important point to note here is the form of the functional (2.2). Namely, we consider -invariant functionals whose gradients are of the form of compact perturbation of the identity. In Theorem 2.4 we have formulated sufficient conditions for the existence of global bifurcations of -orbits of critical points of -invariant functionals. Sufficient condition for the existence of local bifurcation of critical -orbits has been presented in Theorem 2.6. In Remarks 2.5 and 2.7, we have reformulated assumptions of Theorems 2.4 and 2.6, respectively, to make them easier to understand.

In Section 3 we study bifurcations of nonconstant solutions of the Ginzburg-Landau bifurcating from the set of constant solutions. In the first part of this section we show that the functional corresponding to the Ginzburg-Landau equation satisfies all the assumptions of the functional considered in Section 2. We consider two cases of the Ginzburg-Landau equation.

In Section 3.1 we assume that the magnetic field vanishes. In Lemma 3.2 we have described the set of constant solutions of the Ginzburg-Landau system (3.7), which consists of two families. Moreover, we have proved the necessary condition for the existence of bifurcation of nonconstant solutions from these families. The sets of local and global bifurcation points of nonconstant solutions of system (3.7) have been described in Theorem 3.3.

In Section 3.2 we assume that the norm of the magnetic field is constant and different from . Without loss of generality we assume that this norm is equal to . The structure of this subsection is similar to that of Section 3.1. In Lemma 3.4 we have described the set of constant solutions of system (3.2), which consists of three families. Moreover, we have proved sufficient conditions for the existence of local and global bifurcations of nonconstant solutions of system (3.2) from these families. The necessary conditions for the existence of local and global bifurcation of nonconstant solutions of system (3.2) from the families of constant solutions have been proved in Theorem 3.5.

In Section 4 we have shown that we cannot use the Leray-Schauder degree and the famous Rabinowitz alternative to study solutions of problem (1.1). Moreover, we have formulated an open question concerning bifurcations of nonconstant solutions of problem (1.1). This question is at present far from being solved. Finally, we have shown that for domains with sufficiently small volume the first eigenvalue of the magnetic Laplace operator equals . This property has allowed us to simplify the formulation of Theorem 3.5, see Corollary 4.2.

In the appendix we have recalled for the convenience of the reader some material on equivariant algebraic topology thus making our presentation self-contained.

2. Bifurcations of Critical Orbits

In this section we summarize without proofs the relevant material on the equivariant bifurcation theory. In the next section we will apply these abstract results to the study of nonconstant solutions of the Ginzburg-Landau equation.

Throughout this paper stands for the group of complex numbers of module . We identify this group with the group of special orthogonal two-dimensional matrices as follows . Consider a real Hilbert space which is an orthogonal -representation. The -action on the space we define by . For define the orbit of by and the isotropy group of by . Assume that . Hence, is a manifold such that A functional is called -invariant provided that for every and . The space of -invariant functionals of the class will be denoted by . An operator is said to be -equivariant if for every and . It is a known fact that if then , where is the gradient of with respect to the first coordinate. Note that if , then the gradient vanishes on the orbit .

Fix . It is of our interest to study solutions of the following equation:

We are going to apply the bifurcation technique for -orbits of critical points of -invariant functionals. More precisely, we will apply the -equivariant Conley index [11, 12] and the degree for -equivariant gradient maps [16, 17] to prove a local and a global bifurcation of critical -orbits of problem (2.1).

Fix and such that for . For define a connected family , where and . Finally define and assume that . The set is called the set of trivial solutions of problem (2.1). Define . Fix and denote by , a connected component of such that .

Definition 2.1. A point is said to be a local bifurcation point of solutions of (2.1), if . The set of local bifurcation points will be denoted by . A point is said to be a global bifurcation point of solutions of (2.1), if either or is unbounded. The set of global bifurcation points will be denoted by .

In other words a point is a bifurcation point of nontrivial solutions of (2.1) provided that it is an accumulation point of nontrivial solutions of this equation. A bifurcation point is a global bifurcation point of nontrivial solutions of (2.1) provided that a connected set of nontrivial solutions of (2.1) bifurcating from this point satisfies the Rabinowitz-type alternative, that is, either is unbounded or meets the set at least at two times.

Remark 2.2. Directly from the above definition it follows that . Moreover, if (), then () for every .

From now on we assume that the functional is of the following form: where is a compact operator.

The natural question is the following: what is the necessary condition for the existence of bifurcation points of solutions of (2.1)?

In the lemma below we answer the above-stated question.

Lemma 2.3. If , then .

Fix and such that there is satisfying the following conditions:(1), (2)if and , then .

Since is a nondegenerate critical -orbit of the functional , there is an open bounded -invariant subset satisfying . Under these assumptions one can compute the index of an isolated critical orbit in terms of the degree for -equivariant gradient maps, see [13, 15–17], that is, , where is the Euler ring of the group , see [19, 20]. For the convenience of the reader, one has reminded the definition of the Euler ring of the groups in appendix.

It is a known fact that change of any reasonable degree along the set of trivial solutions implies a global bifurcation of zeroes. In the theorem below we formulate sufficient condition for the existence of a global bifurcation of critical -orbits of problem (2.1). The proof of the following theorem is standard and therefore we omit it.

Theorem 2.4. If , then .

Finite-dimensional equivariant Conley index has been considered in [12]. Infinite-dimensional generalisation of equivariant Conley index one can find in [11]. In this paper one considers the -equivariant Conley index defined by a flow induces by .

Assume additionally that for every . Since is isolated in , it is an isolated invariant set in the sense of the -equivariant Conley index theory.

It is a known fact, see Lemma 5.7 of [16], that is an -CW-complex which consists of a base point and one -cell of dimension and of isotropy group , where is the Morse index.

Remark 2.5. Note that where is the -equivariant Euler characteristic, see [19, 20]. The assumption of the above theorem is a little bit mysterious. Taking into account the above, it can be equivalently formulated in the following way: is odd. We underline that since is of the form of compact perturbation of the identity, these Morse indices are finite.

A finite-dimensional version of the following theorem has been proved in [12]. We can literally repeat this proof replacing the finite-dimensional -equivariant Conley index by its infinite-dimensional generalization.

Theorem 2.6. If , then .

Remark 2.7. Similarly as in the case of Theorem 2.4 one can reformulate the assumption of the above theorem. Equivalent but easier to understand formulation is the following:

3. Results

In this section we prove the main results of our paper. Namely, we study bifurcations of nonconstant solutions of the following Ginzburg-Landau equation: where is an open, bounded domain with smooth boundary , is an outward normal to at and .

After making in problem (3.1) a linear transformation for we obtain an equivalent system of real equations

We are going to consider solutions of system (3.2) as critical points of -invariant functional of the class .

Define scalar products as follows:

We underline that norms are equivalent, see [10, 21]. Now define scalar products by where . For simplicity of notation put . For put . Since and , norms are equivalent. From now on we consider as a Hilbert space with the scalar product . It is easy to check that is an orthogonal -representation with -action given by , where is the transposition of . Define a map by and a functional as follows: where .

Remark 3.1. It is easy to verify that . Indeed, this is a standard fact that the functional is of the class . What is left is to show that the functional is -invariant. For put and note that . Moreover, it is clear that for we have , where (1) is a linear, compact, bounded, self-adjoint, positively definite, and -equivariant operator,(2) is a -invariant functional of the class such that(a) is a compact, -equivariant operator,(b), (c) at , uniformly on bounded -intervals.
It is easy to verify that the gradient is an -equivariant operator of the class of the form . Fix , and note that the study of is equivalent to the study of solutions of the following system: where .
It is a known fact that -orbits of solutions of system (3.2) are in one to one correspondence with the critical -orbits of the functional , that is, with the -orbits of solutions of the following equation:
From now on we study bifurcations of solutions of the above equation. For simplicity of notations we write instead of . Let denote the set of eigenvalues of the following eigenvalue problem It is known that if and only if , see [8]. It is clear that .

3.1. Case

In this section we study bifurcations of solutions of the simplified Ginzburg-Landau equation, that is, we assume that the magnetic field vanishes. Such an equation has been considered in [22]. To underline that we write instead . Note that for , where is the usual Laplace operator on . Note that (with Neumann boundary data). If , then denotes the eigenspace of corresponding to . Finally put .

Since , system (3.2) has the following form

The simplest solutions of system (3.7) are constant solutions. We call them trivial and denote . The set of trivial solutions consists of two families and . In the lemma below we have described the set of trivial solutions of system (3.7) and have proved the necessary conditions for the existence of bifurcation points of solutions of this problem.

Lemma 3.2. Under the above assumptions, where and . Moreover,(1)if , then and ,(2)if , then .

Proof. It is easy to check that the set of constant solutions of problem (3.7) is equal .(1)By Lemma 2.3, if , then . Putting in (3.5) we obtain which completes the proof.(2)By Lemma 2.3, if , then . Putting in (3.5) we obtain which completes the proof.

In the theorem below we study local and global bifurcations of nonconstant solutions of system (3.7) from the set of constant solutions. The idea of proof is natural. We compute the -equivariant Conley index and the degree for -equivariant gradient maps along the families of constant solutions and determine levels at which these invariants change.

Theorem 3.3. Under the above assumptions,(a)(b)
Moreover, if , then the continuum is either unbounded or .

Proof. Fix an arbitrary . By Lemma 3.2 we have . It is simple matter to check that is a nontrivial even-dimensional orthogonal -representation. Combining Theorems 4.5 and 4.7 of [20] we obtain . If and , then applying Theorem 4.7 of [20] we obtain that the continuum is unbounded.
Choose such that
To shorten notation set , , , and define . It is understood that .
We claim that(c1), (c2) is negatively defined on ,(c3) is negatively defined on .
Indeed, like in the proof of Lemma 3.2 the study of is equivalent to the study of solutions of the following system:
From condition (3.11) it follows that the linear space of solutions of (3.12) is spanned by . Hence, . Examining (3.12) we obtain that(1)the operator is negatively defined on the space ,(2)the operator is negatively defined on the space , which completes the proof of (c1), (c2), and (c3).
Since , there is an open, bounded, and -invariant subset such that . Moreover, since the isotropy group of every is trivial, we obtain(1), (2).
Taking into account that and the above we obtain
From the above it follows that
Hence, by Theorem 2.4 we obtain provided that , which completes the proof of (b). Fix and note that(i)the Conley index is an -CW-complex with -CW-decomposition ,,(ii)the Conley index is an -CW-complex with -decomposition , .
Note that , where is the -equivariant Euler characteristic, see [19, 20].
Since the Conley indices are not -homotopically equivalent, applying Theorem 2.6 we obtain , which completes the proof of (a).

3.2. Case

In this section we study bifurcations of solutions of the Ginzburg-Landau equation (3.1) assuming that for every .

In the lemma below we have described the set of constant solutions of system (3.2) and have proved the necessary conditions for the existence of bifurcation points of solutions of this problem. To simplify notation we put .

Lemma 3.4. Under the above assumptions, where .
Moreover,(1)if , then ,(2)if and , then and ,(3)if and , then .

Proof. First of all we are looking for constant solutions of system (3.2). Put in (3.2). Taking into account that we obtain system (3.2) in the following form Solving this system we obtain the following set of solutions: which completes the proof.(1)Applying Lemma 2.3 we obtain that if , then . Putting in (3.5) we obtain the following system . Since , the proof is completed.(2)Fix . Putting in (3.5) we obtain the following system . Note that the vector solves this system. Hence, . By Lemma 2.3, if , then . Since , if and only if and which completes the proof.(3)Fix . Putting in (3.5) we obtain the following system . Note that the vector solves this system, that is, . By Lemma 2.3, if , then . Since , if and only if , which completes the proof.

Put . In the theorem below we study local and global bifurcations of nonconstant solutions of Ginzburg-Landau equation (3.2). The proof is similar in spirit to that of Theorem 3.3. For simplicity of notation put and .

Theorem 3.5. Under the above assumptions,(a)(b)
Moreover, for every (1), (2)either the continuum is unbounded or .

Proof. Bifurcations from the family . Note that is a nontrivial even-dimensional orthogonal -representation for every . Combining Theorems 4.5 and 4.7 of [20] we obtain and for every the continuum is unbounded or . Taking into account that is an isomorphism we obtain .
Bifurcations from the family . Fix an arbitrary and choose such that
For brevity set , , , , , and . It is understood that and provided that . We claim that(c1), (c2) is negatively defined on the space of dimension ,(c3) is negatively defined on the space of dimension .
Indeed, like in the proof of Lemma 3.4 the study of is equivalent to the study of solutions of the following system:
From condition (3.19) it follows that the linear space of solutions of (3.20) is spanned by . Hence, . Fix and . Then (3.20) is equivalent to the following system That is why(1)the operator is negatively definite if and only if and ,(2)the operator is negatively definite if and only if and ,which completes the proof of (c1), (c2), and (c3).
Since , there is an open, bounded, and -invariant subset such that . Moreover, since the isotropy group of every is trivial, we obtain(1), (2).
Taking into account that and the above we obtain
From the above it follows that
Hence, by Theorem 2.4 we obtain , provided that . Fix and note that (i)the Conley index is an -complex with -decomposition ,(ii)the Conley index is an -complex with the following -CW-decomposition .
Since the Conley indices are not -homotopically equivalent, applying Theorem 2.6 we obtain .
Bifurcations from the family . This case can be handled in much the same way as bifurcations from the family .

4. Remarks and Open Questions

We can substantially simplify Theorem 3.5 assuming that the first eigenvalue of the magnetic Laplace operator is equal to . In the lemma below we show that for a domain of sufficiently small volume this assumption is satisfied.

Lemma 4.1. Assume that . Then there is such that if , then the first eigenvalue of the magnetic Laplace operator is equal to , that is, .

Proof. Let us consider an eigenvalue problem . Putting we obtain equivalent system of the form
Multiplying equations by , respectively, and integrating by parts we obtain the following equality:
We remind that . To complete the proof it is enough to show that for every the following inequality holds true:
Let be the first positive eigenvalue of the Laplace operator under the Neumann boundary condition. Taking into account that we obtain the following estimation: where is the norm. By the Cheeger's inequality we have as , see [23], that is, for any exists such that if , then . To complete the proof it is enough to choose for .

Combining Lemmas 3.4 and 4.1 with Theorem 3.5 we obtain the following corollary. In this corollary we use the notation of Theorem 3.5. Since , the sets are less complicated. What is important is that we obtain unbounded continua , that is, we have excluded one possibility of behavior of bifurcating continua in the famous Rabinowitz alternative.

Corollary 4.2. Choose a domain such that , where is given by Lemma 4.1. Then and . By Lemma 3.4 one obtains . Applying Theorem 3.5 one obtains(a)(b)
Moreover, for every the continuum is unbounded.

Note that in the proofs of Theorems 3.3 and 3.5 one cannot replace the degree for -equivariant gradient maps by the Leray-Schauder degree. Indeed, since , , see [24, 25]. The famous Rabinowitz alternative, see [26], is not applicable in our paper because is even. Moreover, if the multiplicity is even, then one cannot apply the degree for -equivariant gradient maps to prove the existence of a global bifurcation, see assumption (3.14) (see assumption (3.22)). On the other hand, to prove the existence of a local bifurcation one applies the -equivariant Conley index.

It is a known fact that change of the Conley index implies only a local bifurcation of critical points, see [27–30] for examples and discussion. Therefore, it is natural to rise the following question. Fix and .

Is it true that ?

This question is at present far from being solved.

From now on one replaces the group with an arbitrary compact connected Lie group and assume that is of the form , where(1) is a linear, compact, bounded, self-adjoint, positively definite, and -equivariant operator,(2) is a -invariant functional of the class such that(a) is a compact, -equivariant operator,(b), (c) at , uniformly on bounded -intervals.

The gradient is a -equivariant operator of the class of the form . It is clear that for any . Define . Following [31] one can introduce a notion of nonlinear eigenvalue of .

Definition 4.3. is a nonlinear eigenvalue of , if is a bifurcation point of solutions of equation from the curve for any satisfying the above conditions.
Forgetting about variational structure and -symmetries one can study bifurcations of solutions of the equation applying the Leray-Schauder degree. It is clear that is a nonlinear eigenvalue of if and only if is an eigenvalue of of odd multiplicity, see, for instance, Theorem 1.2.1 of [31].
Computing the bifurcation index in term of the degree for -equivariant gradient maps we obtain the following theorem, see Theorem 4.5 of [20].

Theorem 4.4. is a nonlinear eigenvalue of if and only if is an eigenvalue of odd multiplicity or the eigenspace of corresponding to the eigenvalue is a nontrivial -representation.

In this paper . All the eigenvalues of are of even multiplicity and corresponding eigenspaces of are nontrivial -representations. Summing up, inverse of any eigenvalue of is a nonlinear eigenvalue of . Therefore, one obtains global bifurcations of critical -orbits from the family in Theorems 3.3 and 3.5. Note that if is of sufficiently small volume, then continua bifurcating from family are unbounded, see Corollary 4.2.

Finally we underline that we have proved local and global bifurcations of critical -orbits from connected sets of trivial solutions which are not of the form , see local and global bifurcations from families in Theorems 3.3 and 3.5.

Appendix

In this section for the convenience of the reader we repeat the relevant material from [19], without proofs, thus making our exposition self-contained.

Let , and . For define a -action by . The set of orbits will be denoted .

Definition A.1. Let be a pair of compact -spaces and let . We say that is obtained from by attaching the family of equivariant -cells of orbit type if there exists a -equivariant map which maps -homeomorphically on .

Definition A.2. Let be a pointed compact -space. If there is a finite sequence of -spaces such that(1), (2), where ,(3) is obtained from by attaching a family of equivariant -cells of orbit type , for ,then -space is said to be a finite --complex. The set of subspaces is said to be the cell decomposition of and the set is said to be the orbit type of the cell decomposition of .
We denote by the full subcategory of whose objects are finite --complexes and by subset of consisting of -homotopy types of finite --complexes. Let be the free abelian group generated by the pointed -homotopy types of finite --complexes and let be the subgroup of generated by all elements for pointed --subcomplexes of a pointed --complex .

Definition A.3. Put and let be the class of in . The element is said to be an -equivariant Euler characteristic of a pointed --complex .
For let denote an -homotopy type of the wedge . Since ,
For let . The assignment induces a product given by

Lemma A.4. with an additive and multiplicative structures given by (A.2) and (A.3), respectively, is a commutative ring with unit .

One calls the Euler ring of . If is a --complex without base point, then by one denotes pointed --complex and consequently one puts .

Lemma A.5. is the free abelian group with basis . Moreover, if and is the orbit type of the cell decomposition of , then , where and is the number of -dimensional cells of orbit type .

Below we present additive and multiplicative structures in the Euler ring . It is easy to see that for , the following holds true(1), (2). That is why if , then(1), (2), (3), (4).

Acknowledgments

This work has been partially supported by the National Science Center, Poland, and by the Japan Society for the Promotion of Science Fellowship L-10515.

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Copyright © 2012 Norimichi Hirano and Sławomir Rybicki. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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