Nonlinear Stability of Periodic Traveling Wave Solutions for -Dimensional Coupled Nonlinear Klein-Gordon Equations

Abstract

We study the existence and orbital stability of smooth periodic traveling waves solutions of the -dimensional coupled nonlinear Klein-Gordon equations. Such a system occurs in quantum mechanics, fluid mechanics, and optical fiber communication. Inspired by Angulo Pava’s results (2007), and by applying the stability theory established by Grillakis et al. (1987), we prove the existence of periodic traveling waves solutions and obtain the orbital stability of the solutions to this system.

1. Introduction

This paper pays close attention to -dimensional system of coupled nonlinear Klein-Gordon equations where is Laplace operator, , , , (), , and and are complex functions.

Equation (1) frequently describes physical questions, for example, crystal growths and dislocations. In recent decades, lots of methods have been put forward, such as the direct integral method, the hyperbolic function expansion method [1, 2], mixing exponential method, and Jacobi elliptic function expansion method [3, 4], to obtain the exact solutions of nonlinear evolution equations.

In this paper, we are interested in the existence of a smooth periodic solution and the orbital stability of solution of (1). A comprehensive development of stability results for this type has been acquired by Grillakis et al. [5, 6], Benjamin [7], and Weinstein [8]. Moreover, many results of orbital stability of Klein-Gordon equation have been attained. For example, Shatah [9] has given sufficient conditions of orbital stability in such a case: and it is showed that the standing wave solutions of this equation have orbital stability when , with as an integer. In addition, Ohta and Todorova [10] have established the instability of (2) when . In this case, we say that this wave is unstable for (2); suppose for any , there exists such that and the solution of (2) with satisfies . Moreover, Angulo Pava [11] has studied the orbital stability of cnoidal waves solution of the following system: where , and he obtained the orbital stability of the cnoidal solutions in the energy space with regard to the periodic flow of (3) when .

In our paper, we work on the existence and orbital stability of periodic traveling waves solution of system (1). The plan of the paper is as follows: in Section 2, we obtain the existence of periodic traveling waves solution of system (1). In Section 3, the spectrum of operator (see (26)) is studied in detail. In Section 4, the orbital stability of system (1) is established. In Section 5, we give a brief discussion and conclusion to state the application of system (1) in the field of engineering.

Notation. Let denote Lebesgue measurable space for the open subset in , the norm , and . denotes the complex space with real inner product . The dual space of is . There exists a natural isomorphism denoted by . Here,

2. Existence of the Periodic Traveling Wave Solution for the KG Equations

In this section, motivated by Angulo Pava’s work [12], we will study the existence of a smooth curve of periodic traveling waves solutions to the coupled nonlinear Klein-Gordon equations (1) with the form where , , , and . The profile satisfies the boundary conditions and as and the constant denotes the speed of traveling waves.

By substituting (5) in (1), we obtain where and .

Now, we consider the the special solutions of (6) where . Then, (6) is transformed into where and . Moreover, we assume that and .

Notice the relation ; hence, the corresponding parameters satisfy the relations and .

Next, to study the solutions of (7), we only need to consider the first equation of (7): Firstly, we suppose that and .

Multiplying (8) by and integrating once, we obtain where is a constant of integration, , , and . Without loss of generality, we suppose that . Define and ; hence, (9) becomes

Next, we continue to define a new variable , which satisfies and . By relation , we obtain

So, we get for that Next, inspired by Angulo Pava’s work [12], by applying the theory of Jacobian elliptic function [13], we obtain the dnoidal wave solution of (8), with

Because dn has fundamental period ; that is, . Here is the complete elliptic integral of first kind. Hence, we know that solution (13) with fundamental period is as follows:

Next, expression (15) can be converted into a simple form according to (14). In fact, if we fix variables and , we can find (15) as a function with only one variable :

Furthermore, if , , and , . If , , and , therefore, .

Then, we will establish a smooth dn wave solution for (8) with fixed period . For any given , , and , we can know that the mapping is strictly decreasing function. In what follows, we will show that there is a unique such that is a fundamental period of (13).

Theorem 1. For any given , consider ; then there is a unique such that . Then,(1)there exist an interval around , an interval around , and a smooth and unique function , such that and  Here, and .(2)The solution is determined by and , has fundamental period , and satisfies (13). In addition, the mapping is a smooth function.

Proof. The proof is via the implicit function theorem, the method is based on Angulo Pava’s work [12], and for more details, one may see Theorem 1 in [12].

Theorem 2. Let . Then, there is a smooth curve of periodic traveling wave solutions for (8) with . Here, , .

3. Spectral Analysis

Firstly, (1) can be rewritten as the following form: Moreover, let . So, (1) can be rewritten as the following Hamiltonian system: where is the skew-symmetric linear operator:

Let be a one-parameter group of unitary operator on defined by where and .

Obviously,

Since , we can deduce that

Next, we define the functional

According to [5], it can be verified that and are conserved quantities, and the solutions of system (1) are critical points of ; namely,

Next, we define an operator from to : Then, for any ,

By a simple computation, we can deduce that is a self-adjoint operator. It is shown that is a bounded self-adjoint operator. The spectrum of the consists of the real numbers such that is not invertible, and belongs to the spectrum of .

Moreover, we can also prove that

Let

From (28), we can know that is contained in the kernel of .

By [14], we have the following theorem.

Theorem 3. The space is decomposed as a direct sum; . Here, is defined by (29), is a finite-dimensional subspace, such that and is a closed subspace, such that where is a positive number and independent of .

Proof. For any , let Then, where As , . So that Moreover, we may obtain that From (13) and [5, 15], they imply has a simple zero. By Sturm-Liouville theorem, 0 is the second eigenvalue of , and has exactly one strictly negative eigenvalue , with an eigenfunction ; that is, Furthermore, is the first simple eigenvalue of . From [16], we have the following lemmas (Lemmas 4 and 5).
Lemma 4. For any real function , if it satisfies There exists a number independent of , such that Lemma 5. For any real function , if it satisfies Then, there exits a number independent of , such that Let . By (3), we can obtain Notice that the kernel of is spanned by and ; then Let For any and , we choose , , and . Then can be uniquely represented by This shows that the space is decomposed as a direct sum; .
In the following, we will prove that Let . By Lemmas 4 and 5, we have where . Thus, we complete the proof of Theorem 3.

4. Orbital Stability of the Periodic Wave Solutions for the -Dimensional KG Equations

In this section, we shall apply the method established by Grillakis et al. [5] to give the orbital stability of the periodic traveling solutions of system (1): where

By [16, 17] and by applying classical semigroups method, for any there exists a solution for system (1) and or .

In Section 2, we have obtained that system (1) has periodic traveling waves solution of (48) in . Moreover, in Section 3, we also obtained that operator has only one negative simple eigenvalue, the kernel of is spanned by , and the rest of its spectrum is positive and bounded away from zero. According to orbital stability theory of Grillakis et al. [6], we only need to prove that the function is strictly convex as follows:

Firstly, we state our definition of orbital stability.

Definition 6. The orbit of is defined by We say orbit is orbital stable in Banach space , if for all there exists a such that if satisfies , and is a solution of (1) in internal with the initial value , then can be extended to a solution in has Otherwise, we say that is orbital unstable.

Theorem 7. Let and , determined by (49), be a periodic traveling waves solution of system (1). Then, this solution is orbital stable in by periodic flow of system (1).

Proof. From (3), we know that Therefore Then Through (13), we obtain that Now, by using that (see [13]), Therefore, we can obtain that So that Indeed, So that Moreover, according to [13], is strictly increasing functions; hence, . Hence, we can obtain . So, we complete the proof of this theorem.

5. Discussion and Conclusion

In this paper, inspired by Angulo Pava’s ideas [11, 12, 18, 19], and by using the stability theory established by Grillakis et al. [5], we obtain solution (48) of system (1) and prove that the solution is orbital stable. The method is helpful to look for periodic solution and obtain the orbital stability for a class of nonlinear equations, which is widely used in quantum mechanics, fluid mechanics, and optical fiber communication. Hence, it may contribute to solving these engineering problems.

Conflict of Interests

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