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Abstract and Applied Analysis

Volume 2014, Article ID 649270, 15 pages

http://dx.doi.org/10.1155/2014/649270
Research Article

Quasiperiodic Solutions of Completely Resonant Wave Equations with Quasiperiodic Forced Terms

1School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China

2Key Laboratory of Symbolic Computation and Knowledge Engineering, Ministry of Education, Jilin University, Changchun, Jilin 130012, China

3Fundamental Department, Aviation University of Air Force, Changchun 130023, China

Received 28 February 2014; Revised 25 April 2014; Accepted 28 April 2014; Published 18 May 2014

Academic Editor: Yongli Song

Copyright © 2014 Yixian Gao et al. 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.

Abstract

This paper is concerned with the existence of quasiperiodic solutions with two frequencies of completely resonant, quasiperiodically forced nonlinear wave equations subject to periodic spatial boundary conditions. The solutions turn out to be, at the first order, the superposition of traveling waves, traveling in the opposite or the same directions. The proofs are based on the variational Lyapunov-Schmidt reduction and the linking theorem, while the bifurcation equations are solved by variational methods.

1. Introduction

This paper is devoted to the study of the existence of small-amplitude quasiperiodic solutions of completely resonant forced nonlinear wave equations like with periodic boundary conditions while the nonlinear forced term is when the traveling waves are in the same directions, and is when the traveling waves are in the opposite directions. Moreover, the nonlinear forced terms are all analytic in a neighborhood of .

Periodic or quasiperiodic solutions in nonresonant PDEs have been obtained, for instance, in [112] by the Lyapunov-Schmidt reduction together with Nash-Moser theory and KAM theory, while the completely resonant autonomous PDEs have been originally studied by variational methods starting from Rabinowitz [1319]. They obtained the existence of periodic solutions with period being a rational multiple of , and such solutions correspond to a zero-measure set of values of the amplitudes. The case with period being irrational of , which in principle could provide a large measure of values, has been mostly studied under strong Diophantine conditions; see [2025] and the references therein. In [26, 27], using the Lindstedt series method, Gentile and Procesi obtained the existence of periodic solutions for a large measure set of frequencies for the nonlinear wave equations and nonlinear Schrödinger equations with periodic boundary conditions. In [28], Yuan obtained the existence of quasiperiodic solution for a large measure set of at least three dimensional rotation vectors by the KAM method. In [29], under the periodic boundary condition and with the periodic forced nonlinearities , Berti and Procesi got the existence of quasiperiodic solution of nonlinear wave equation in the form of . In [30], Procesi firstly obtained the quasiperiodic solutions with two frequencies in the form of for the specific nonlinearities , where the forced terms do not depend on the time and the bifurcation equations are solved by ODE methods. In [31], Baldi proved the existence of small-amplitude quasiperiodic solutions in the form of with the general nonlinearities , which also do not depend on the time. In [32], they considered the existence of quasiperiodic solution of the forced wave equation, in which the solutions are traveling in opposite directions. However, they did not give the regularity of the solutions, and the results are the special case of our results in Section 4. Moreover, we mention the work [33] of Bambusi, where a simple proof of an infinite-dimensional extension of the Lyapunov center theorem is given.

In this paper, for the completely resonant wave equation (1) subjecting to the quasiperiodic forced terms, we will prove the existence and regularity of quasiperiodic solutions with two frequencies, , in both of the following two cases.

Case (A1). The first case considers the wave traveling in the same directions .

Case (A2). The second case considers the wave traveling in opposite directions .

2. Main Results

We look for quasiperiodic solutions of (1) of the following form:(in the same directions) with frequencies , or (in opposite directions) with frequencies , imposing the frequencies to be close to linear frequency 1. Therefore, finding the quasiperiodic solutions of (1) with frequencies, respectively, is equivalent to finding periodic solutions with respect to for the following equations:(in the same directions) (in opposite directions) We assume that the quasiperiodic forced term , is analytic in but has only finite regularity in . More precisely,(H) , and the coefficients verify, for some , . The function is not identically constant in .

We look for solutions of (7)-(8) in the Banach space where denotes its complex conjugate, and .

The space is a Banach algebra with respect to multiplications of functions, namely, We will prove the following theorems.

Theorem 1. Assume that the nonlinearity satisfies (H) and . Let be the uncountable zero-measure Cantor set where , , are sets of badly approximate numbers defined as for , and . There exist constants , and , such that, for , there exists a solution of (7), having the form with where is a constant. As a consequence, (1) possesses the quasiperiodic solutions, traveling in the same directions, , with two frequencies .

Theorem 2. Assume that satisfies assumption (H) and . Let be the uncountable zero-measure Cantor set where is a set of badly approximate numbers defined as for , and . There exist positive numbers , such that, , (8) admits solutions in the form of satisfying As a consequence, (1) possesses the quasiperiodic solutions, , traveling in opposite directions.

Remark 3. The quasiperiodic solutions of traveling waves we obtained are different from the ones got by KAM methods since the quasiperiodic solutions we get depending on and are coupled and in the form of the traveling waves.

Remark 4. We can get the similar result with more general nonlinearity, such as , for any .

This paper is organized as follows: we first prove the existence of quasiperiodic solutions, at the first order, to the superposition of traveling waves, traveling in the same directions. In Section 4, we prove the existence of quasiperiodic solutions traveling in opposite directions.

3. Waves Traveling in the Same Directions

Substituting into (7), we can obtain the equations where, see (7), we have and . To prove Theorem 1, instead of looking for solutions of (7) in a shrinking neighborhood of zero, it is convenient to perform the rescaling , enhancing the relation between the amplitude and the frequencies. Without confusion, we define so the problem becomes To find the solutions of (23), we will apply the Lyapunov-Schmidt reduction method which leads to solving separately a “range equation” and a “bifurcation equation.” In order to solve the range equation (avoiding small divisor problems), we restrict to the uncountable zero-measure set for Theorem 1, and we apply the Contraction Mapping Theorem; similar nonresonance conditions have been employed, for example, in [2123, 25, 29, 30].

Equation (23) is the Euler-Lagrange equation of the action functional defined by where To find critical points of , we perform a variational Lyapunov-Schmidt reduction inspired by Berti and Bolle [22, 23, 29]; see also Ambrosetti and Badiale [34].

3.1. The Variational Lyapunov-Schmidt Reduction

The operator is diagonal defined on the Banach space under the Fourier basis with eigenvalues So, we have The critical points of the unperturbed functional form an infinite-dimensional linear space , and they are the solutions of the equation The space can be written as In view of the variational argument that we will use to solve the bifurcation equation, we split as , where with We will also use in the norm So, we can decompose the space , where Projecting (23) onto the closed subspaces and , setting with and , we obtain where are the projectors, respectively, onto and .

In order to prove analyticity of the solutions and to highlight the compactness of the problem, we perform a finite-dimensional Lyapunov-Schmidt reduction, introducing the decomposition , where Setting with and , we finally get where are the projectors onto    , and is the projector onto . We will solve first the - -equations for all , provided belongs to a suitable Cantor-like set, is sufficiently small, and is large enough (see Lemma 7). Next, we will solve the -equation by means of a variational linking argument; see Section 3.4.

3.2. The - -Equations

We first prove that restricted to has a bounded inverse when belongs to the uncountable zero-measure set where , , is a set of badly approximate numbers defined by for , and . , , accumulate at and zero, respectively, from both the right and the left; see [21, 31, 35].

The operator is diagonal in the Fourier basis with eigenvalues

Lemma 5. For , the eigenvalues of restricted to satisfy As a consequence, the operator has a bounded inverse and satisfies

Proof. Denote by the nearest integer close to and by its fractional part. If both and , we have If , then , so that . This implies that In the same way, if , we have So, the operator restricted to has a bounded inverse and satisfies

Lemma 6. The operator has a bounded inverse , satisfying

Proof. is diagonal in the Fourier basis of with with eigenvalues The eigenvalues of restricted to verify , where the constant depends on , and (48) holds.

Fixed points of the nonlinear operator defined by are solutions of the - -equations. Using the Contraction Mapping Theorem, we can prove the following lemma.

Lemma 7. For any , there exist an integer and positive constants such that and there exists a unique solution of the - -equations satisfying Moreover, the map is and

Proof. Let us consider the ball with norm . We can claim that, under assumptions (51), there exist such that the map is a contraction mapping in , that is, we have to prove (i) ; (ii) ,

where the constant . In the following,    denote different constants. By (48) and the Banach property of , Similarly, for , by (43), we have For all , setting , we can get whenever . Thus, , we get Now, set , and we define . By the inequality above, there exist and such that and , So, we get the proof of (i). Item (ii) can be obtained with the same estimates.

By the Contraction Mapping Theorem, there exists a unique fixed point of in . The bounds of (52) follow by the definition of and .

Since the map , the Implicit Function Theorem implies that the map is . Differentiating both sides of , we can get Using (43)–(48) and the Banach property of , we get which implies the bounds (53), since when is small enough, we can get

3.3. The -Equation

Once the - -equations have been solved by , there remains the finite-dimensional -equation The geometric interpretation of the construction of is that, on the finite- dimensional submanifold , diffeomorphic to the ball the partial derivatives of the action functional with respect to the variables vanish. We claim that, at a critical point of restricted to , also the partial derivative of with respect to the variable vanishes, and therefore such a point is critical also for the nonrestricted functional .

Actually, the bifurcation equation (63) is the Euler-Lagrange equation of the reduced action functional

Lemma 8. and a critical point of is a solution of the bifurcation (63). Moreover, can be written as where and for some positive constant , we can get

Proof. By (37) and (38), we have that, at , Since and , we deduce that and therefore solves also the -equation (36). Write , where is a homogeneous functional of degree two. By homogeneity, and, according to (69), we have Substituting the above equality into (72), we obtain, at , Because , by (52), we can get the bounds of (68).

The problem of finding nontrivial solutions of the -equation is reduced to finding nontrivial critical points of the reduced action functional in . By (66), this is equivalent to finding critical points of the rescaled functional where the quadratic form is positive definite on , negative definite on , and zero definite on . For and , we have where the positive constants are bounded away from zero and independent of . We will prove the existence of critical point of in of linking type.

3.4. Linking Critical Points of the Reduced Action Functional

We cannot directly apply the linking theorem because is defined only in the ball . Therefore, we first extend to the whole space . We define the extended action function as where is and is a smooth, nonincreasing, cut-off function such that By definition, on and outside . Moreover, by (68), there is a constant such that , and Second, we will verify that satisfies the geometrical hypotheses of the linking theorem.

Lemma 9. There exist positive constants , and , which are independent of , such that, and ,(i) , (ii) ,

where is the rectangle in , and is the unit vector in .

Proof. (i) with , we have Now, we fix small such that