Nonlinear Stability of Periodic Traveling Wave Solutions for ( n + 1 )-Dimensional Coupled Nonlinear Klein-Gordon Equations

We study the existence and orbital stability of smooth periodic traveling waves solutions of the (n + 1)-dimensional coupled nonlinearKlein-Gordon equations. Such a systemoccurs in quantummechanics, fluidmechanics, 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.

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 () arg  = −|| −1 , with  > 1 as an integer.In addition, Ohta and Todorova [10] have established the instability of (2) when 1 + 4/ <  < 1 + 4/( − 2).In this case, we say that this wave   () is unstable for (2); suppose for any  > 0, there exists Pava [11] has studied the orbital stability of cnoidal waves solution of the following system: where  > 0, and he obtained the orbital stability of the cnoidal solutions in the energy space  1 per (0, ) ×  1 per ([0, ]) with regard to the periodic flow of (3) when  ≥ 1/20.
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.
By a simple computation, we can deduce that  is a selfadjoint operator.It is shown that  −1  is a bounded selfadjoint operator.The spectrum of the  consists of the real numbers  such that − is not invertible, and  = 0 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 || → ∞, ,  → 0. 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  1 , and  1 has exactly one strictly negative eigenvalue − 2  1 , with an eigenfunction ; that is, Furthermore, 0 is the first simple eigenvalue of  2 .From [16], we have the following lemmas (Lemmas 4 and 5).

Lemma 4. For any real function
There exists a number  1 ∈  + independent of  1 , such that Lemma 5.For any real function  2 ∈ (), if it satisfies Then, there exits a number  2 ∈  + independent of  2 , such that Let ⃗  = ( 1 , 0, , 0)  ∈ .By (3), we can obtain Notice that the kernel of  is spanned by  → ] 1 and  → ] 2 ; then Let For any  ∈  and ⃗  ∈ , we choose Then  can be uniquely represented by This shows that the space  is decomposed as a direct sum;  =  +  + .
In Section 2, we have obtained that system (1) has periodic traveling waves solution of (48) in  1 per [0, ] ×  1 per [0, ].Moreover, in Section 3, we also obtained that operator  has only one negative simple eigenvalue, the kernel of  is spanned by   (0)Φ, 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 ⃗ Φ(, ) = (, ) is defined by We say orbit Ω ⃗ Φ is orbital stable in Banach space , if for all  > 0 there exists a  > 0 such that if ⃗ Otherwise, we say that ⃗ Φ(, ) is orbital unstable.

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.