Abstract and Applied Analysis

Volume 2014 (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

Yixian Gao,1,2 Weipeng Zhang,¹ and Jing Chang³

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 [1–12] 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 [13–19]. 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 [20–25] 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 [21–23, 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 . Since , by (84), we can get

(ii) Let

with . For ,

because . Now, by Hölder inequality and orthogonality,

and by (88) we deduce that

Now, we fix large such that , and therefore

Next, setting , we fix large such that

and therefore

Finally, if ,

So, if , we can get .

We introduce the minimax class . According to Proposition 5.9 of [19], the maps of have an important intersection property as follows.

Proposition 10. ( and link with respect to ). Consider

One defines the minimax linking level as follows:

Obviously, by Proposition 10 and Lemma 9,

and, therefore, .

Since , so

where is independent of . By the linking theorem, we deduce the existence of a Palais-Smale sequence at the level , namely,

Third, we will prove that the Palais-Smale sequence (up to subsequence) converges to some nontrivial critical point in some open ball of , where and coincide. Because the space is finite dimensional, we only need to prove that the sequence is bounded.

Lemma 11. There is a constant independent of , such that, for all small enough and large enough, the Palais-Smale sequence is bounded; that is, . So, there exists a subsequence of the P-S sequence that converges to some critical point , and the functional possesses a nontrivial critical point with critical value .

Proof. In the sequel, we will always assume that . Writing , by (81)-(82), we can get

Then, by and ,

By Hölder inequality and orthogonality,

and therefore . In the same way, by Hölder inequality and (82),

By (101) and the above inequalities, using , we conclude that . Estimating analogously, we derive . Finally, we deduce that

So, we can conclude that for a suitable constant , which is independent of . Since is finite dimensional, converges, up to subsequence, to some critical point of with . Since , we conclude that .

Proof of Theorem 1. Let us fix and take . According to Lemma 7, we can get, for , a solution of the --equations with . By Lemma 11, the extended functional possesses a nontrivial critical point with . Since coincides with on the ball by Lemma 8, there exists a nontrivial weak solution of (23). Finally, solves (7).

According to Lemma 6, by the regularizing property of the operator , the solution of the -equation belongs to . By the -equation

where , we can get that

for , satisfying . For , the eigenvalues of operator restricted to satisfy , , and, thus, we deduce that , for all (satisfying ), and . By (105),

we get , and , with . Thus, (15) follows with , . By (5), is the solution of (1), for all . To show that is quasiperiodic, it remains to prove that depends on both variables independently. According to Lemma 9, . On the other hand, , , so that , and, therefore, depends on . In fact, any solution of (23) depending only on , that is, the solutions of

is . Indeed, by the homogeneity of , we have

Now consider a smooth function with zero mean, ant it satisfies for some . Multiplying (108) by and integrating over , we have

According to that has zero mean and multiplying above equation by , we get

The function , with , is a nontrivial analytic function. Thus, the equation cannot have a sequence of zeros accumulating to zero. So, for small enough, .

4. Waves Traveling in Opposite Directions

Substituting into (8), we get

where (see (8))

We rescale (112) in order to highlight the relationship between the amplitude and the variation in frequency: , and, for convenience, we assume . In the following, we consider the scaled equation

where .

Equation (114) is the Euler-Lagrange equation of the Lagrange action functional defined by

where and ,

In order to find critical points of , we use the same method as in Section 3. The operator is diagonal defined on the Banach space under the Fourier basis with eigenvalue . So, we have

The unperturbed functional possesses an infinite-dimensional linear space of critical points, which are the solutions of the equation

The space can be written as

We split as

where

We decompose the space , where

Projecting (114) onto the closed subspaces and , setting with and , we obtain

where , are the projectors, respectively, onto and ; moreover, they are continuous.

In the same way, we decompose the space . Setting with and , we finally get

The operator is diagonal in the Fourier basis with eigenvalues . 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 as

for , and .

Lemma 12. 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 . If both and , then we have

If , then

In the same way, if , then we have

Lemma 13. The operator has a bounded inverse which satisfies

Proof. is diagonal in the Fourier basis of with with eigenvalues

The eigenvalues of restricted to verify , where the constant depends on , and (134) holds.

Similarly, the solution of -equation (124) is the Euler-Lagrange equation of the reduced Lagrangian action functional:

Lemma 14. and a critical point of is a solution of the bifurcation equation (124). Moreover, can be written as

where

and, for some positive constant , we can get

The problem of finding nontrivial solutions of the -equation is reduced to finding nontrivial critical points of the reduced action functional in . By (137), this is equivalent to finding critical points of the rescaled functional denoted by and called the reduced action functional

where the quadratic form

is positive definite on , negative definite on , and zero definite on . For , we have

where the positive constants are bounded away from zero and independent of . The following steps of finding the nontrivial solutions of the -equation are similar to Lemmas 9 and 11 in Section 3.4.

Proof of Theorem 2. We can get the solution of (8) as follows:

According to Lemma 13, by the regularizing property of the operator , the solution of the -equation belongs to . By the -equation

where , we can get that

for , satisfying . For , the eigenvalues of operator

restricted to satisfy

and, thus, we deduce that , for all (satisfying ), and . By (144),

we get , and , with . Thus, (19) follows with . By (6), is the solution of (1), for all . Obviously, depends on both variables independently. So, is a quasiperiodic solution of (1), with frequencies .

5. Conclusion

In this paper, for the completely resonant nonlinear wave equations, under periodic boundary conditions, we obtain the existence and regularity of quasiperiodic solutions. The forced terms we consider are quasiperiodic, and, according to the linking theorem, the bifurcation equations are solved by variational method. Moreover, the solutions depending on the the spatial and time variables are coupled and in the form of traveling waves. In [28], Yuan got the existence of quasiperiodic solutions with    frequencies by KAM theory, in which the form of the solutions is . In the future work, we will investigate the existence of quasiperiodic solutions with the traveling wave form as .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors express their sincere thanks to Professor Yong Li for his instructions and many invaluable suggestions. The first author was partially supported by NSFC Grant (nos. 11001042, 11171056, 11171130, and 11271062) and SRFDP Grant (no. 20100043120001). The second author is supported by NSFC Grant (nos. 11001041, 11201360, 11101170, 11202192, and 10926105), SRFDP Grant (no. 200802001008), and the State Scholarship Fund of the China Scholarship Council (nos. 2011662521 and 2011842509).

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