Research Article | Open Access
Nonlinear Stability of Periodic Traveling Wave Solutions for -Dimensional Coupled Nonlinear Klein-Gordon Equations
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.
This paper pays close attention to -dimensional system of coupled nonlinear Klein-Gordon equationswhere 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 , and Weinstein . Moreover, many results of orbital stability of Klein-Gordon equation have been attained. For example, Shatah  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  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  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 , we will study the existence of a smooth curve of periodic traveling waves solutions to the coupled nonlinear Klein-Gordon equations (1) with the formwhere , , , and . The profile satisfies the boundary conditions and as and the constant denotes the speed of traveling waves.
Notice the relation ; hence, the corresponding parameters satisfy the relations and .
Next, we continue to define a new variable , which satisfies and . By relation , we obtain
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:
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.
Theorem 2. Let . Then, there is a smooth curve of periodic traveling wave solutions for (8) with . Here, , .
3. Spectral Analysis
Let be a one-parameter group of unitary operator on defined by where and .
Since , we can deduce that
Next, we define the functional
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
From (28), we can know that is contained in the kernel of .
By , 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 , we have the following lemmas (Lemmas 4 and 5).
Lemma 4. For any real function , if it satisfiesThere 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 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. , 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.
Proof. From (3), we know that Therefore Then Through (13), we obtain that Now, by using that (see ), Therefore, we can obtain that So thatIndeed,So thatMoreover, according to , 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. , 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.
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