Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 427487, 8 pages
http://dx.doi.org/10.1155/2015/427487

Research Article

On the Existence and Stability of Standing Waves for 2-Coupled Nonlinear Fractional Schrödinger System

Xiuyan Sha, Huanmin Ge, and Jie Xin

School of Mathematics and Statistics Science, Ludong University, Yantai 264025, China

Received 25 September 2015; Accepted 1 December 2015

Academic Editor: Seenith Sivasundaram

Copyright © 2015 Xiuyan Sha 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.

We study a system of 2-coupled nonlinear fractional Schrödinger equations. Firstly, we construct constrained minimization problem to the system. Next, we prove the existence of standing waves for the system by using the concentration-compactness and commutator estimates method. Lastly, we also consider the set of minimizers of the constrained minimization problem. We prove that it is a stable set for initial value of the problem; that is, a solution to the system with initial value which is near the set will remain near it for all time.

1. Introduction

In the recent years, more and more researchers study the application of fractional calculus and fractional integrodifferential equations in physics and other areas (see [1–4]). The concept of fractional calculus is firstly put forward by Leibniz as a generalization of standard calculus. Afterwards several kinds of definitions of them have been established such as Riemann-Liouville fractional derivative, Caputo fractional derivative, and Weyl fractional derivative (see [5, 6]). We all know that a variety of fractional calculus is brought in inspired by standard calculus, but the calculation laws of fractional calculus are much different (see [5, 7]). To avoid the complicated fractional calculus, in this paper we are only concerned with the fractional Laplacian operator.

Here we study the existence and stability of standing waves for the following system of -coupled nonlinear fractional Schrödinger equations:

where , , , , , and is a pseudo-differential operator which is defined by , the Fourier transform .

It is well known that the coupled nonlinear Schrödinger equations play an important role in describing nonrelativistic quantum mechanical behavior. In particular, the fractional Schrödinger equations can describe better some real physical phenomenon. Schrödinger type equation(s) is first derived by Feynman and Hibbs, applying path integrals over Brownian paths in [8]. In [9–11], Laskin showed the path integral over Lévy-like quantum mechanical paths approaches to a generalization of quantum mechanics. As the path integral over Brownian trajectories, it leads to the standard Schrödinger equations. And as the path integral over Lévy-like quantum mechanical paths, it leads to the fractional Schrödinger equation. Recently, there are some researchers studying fractional Schrödinger equation(s) and its physical applications (see [12–14]). The research of the existence and stability of standing waves for the nonlinear Schrödinger equations arises in various fields of physics such as plasma physics, constructive field theory, and nonlinear optics (see [15–22]). There are many papers about this topic for the standard Schrödinger equations with different nonlinear terms such as [17, 19, 20]. There are some results concerned with the existence and stability of standing waves for the nonlinear fractional Schrödinger equations in [21, 22]. The author [23] obtained the global solution for a class of systems of fractional nonlinear Schrödinger equations with boundary condition. However, there are few results about the standing waves of fractional nonlinear Schrödinger equations (1). In the present paper, we consider the existence and stability of standing waves for the system of -coupled nonlinear fractional Schrödinger equations.

The rest of this paper is arranged as follows. In Section 2, we recall some basic definitions and introduce preparation results. In Section 3, we give the main results and proof of them.

2. Preliminaries

In this section, we firstly introduce the constrained minimization problem in order to study the existence and stability of standing waves for (1). We know that the standing waves of (1) have the special form , where . Then we only need to find satisfying the following equations to study the existence and stability of standing waves of (1):

where are complex value functions.

By the variational method, we know that in order to study the existence of solution for (2) we need to consider the following constrained minimization problem:

where

And is the fractional order Sobolev space. We denote

as the set of minimizers of the problem .

Next we recall some basic definitions. Define the fractional order Sobolev space

with the norm

Denote the product space with the norm . Throughout this paper, we denote the norm of by for all . Denote the set . For simplicity and convenience, the letter will represent a constant, which may be different from one to others. , for example, represents the constant which can be expressed by the parameters appearing in the braces.

Lastly, we give the following some lemmas, which will be used in our paper. For more detail, we can see [21, 23, 24].

Lemma 1. If , , and , representing the Schwartz class, then the fractional Laplacian of is also expressed for the formula

where means the Cauchy principal value on the integral and is some positive normalization constant.

Lemma 2. For , there are two properties with :

(i)The norm of is equivalent to (ii), ,

which implies , where is defined by the method of trace interpolation (see chapter of [24]) and its norm is given by . and are positive constants.

Remark 3. In the following paper, we denote the norm of by .

Lemma 4 (see [24]). For , , and . Moreover, if in , then in , and almost everywhere in , where .

Lemma 5. If is bounded in and for some , then in for .

Remark 6. The proof of Lemma 5 is obtained easily according to Lemma 2.4 (see [21]), so we will omit it.

Lemma 7 (commutator estimates). If , , the Schwartz class, then

where satisfying .

In [23], the authors have obtained the existence and uniqueness of the global solution of a class of systems for fractional nonlinear Schrödinger equations with periodic boundary condition by using the Faedo-Galërkin method. In the proof of Theorem 4.1 in the references, only needing to let the period , we can obtain the following theorem.

Theorem 8. Let . If then there exists a global weak solution to the initial value problem of coupled nonlinear fractional Schrödinger equations (1) with initial date .

3. Uniform Estimates of the Solution

In the section, we consider coupled nonlinear equations (2). Since the scaling , we can transform (2) to the following equations:

So we only consider coupled nonlinear equations (10). So we have

For any , let and .

Under the case we have the following main results.

Theorem 9. Assuming is a minimizing sequence of problem , then there exists a sequence such that contains a convergent subsequence in . In particular, there exists a minimizer for problem , and

Theorem 10. The set is -stable with respect to (1); that is, , there exists such that if is a solution of (1) with initial value satisfying

then, ,

Before we give the proof of Theorems 9 and 10, we firstly prove the following some lemmas.

Lemma 11. For any , one has the following:

(i),(ii)every minimizing sequence problem is bounded in . And for sufficiently large , there exists a positive constant such that

Proof. (i) For given , , . And set ; then .
By Lemma 1, we have that

By , there exists taking sufficiently small . Therefore, we obtain that .
By the Hölder inequality and the Sobolev inequality, we can derive

where . Because of the Young inequality and , for , from the above inequality we can derive that

So for , setting , by the Young inequality and (17), we have that

Therefore for sufficiently small , we have that

So .
(ii) Let be minimizing sequence for problem .
By (16), we have that

By , we get that ; that is, is bounded in .
Using an argument by contradiction, we prove the second part of (ii). Assuming does not exist such that , then

Now we have that

which contradicts . Therefore for sufficiently large , there exists a positive constant such that We have completed the proof of the lemma.

Now, for minimizing sequence of problem we introduce the Lévy concentration function . Apparently, each function is nondecreasing on . Using Helly’s selection theorem, has a convergent subsequence , such that , for all , where is a nondecreasing function. By , there exists satisfying

Next, we will prove .

Lemma 12. Assuming and is a minimizing sequence for , then .

Proof. By the definition of , we know . Using an argument by contradiction, assume ; then there exist and , satisfying

By Lemma 5, we have in . According to Lemma 11(ii), in . They are contradictory. Therefore, we obtain .

Lemma 13. For , then .

Proof. For given , , let , where , . Then and

where we apply Lemma 1.
Now, we can derive that

Therefore by Lemma 11(ii) and , we have that

Lemma 14. Assuming and , then .

Proof. For given minimizing sequence for problem , , define . Then is nondecreasing on ; moreover for all , has subsequences denoted again by , existing nondecreasing function such that . Therefore such that satisfy

Apparently, there exists , satisfying

where because of definition of and (27).
Setting , they satisfy that ; ; for , ; and . Then we have that

Therefore (29) imply that there exist and , , satisfying , . Then we have that

Using Lemma 7, we derive that

where . By , the Sobolev inequality, Lemma 11, and (31), we obtain that

for sufficiently large . Similarly, for sufficiently large we have that

So we have that

By Lemma 11, (16), (27), , and the definitions of , (), we can derive that

where and . Therefore by the two inequations, we obtain that

By (30) and (36), we obtain

So, we prove that .

Obviously, using Lemmas 11–14, we obtain .

Proof of Theorem 9. From the above parts; we know that , that is, there exists a subsequence of satisfying

Therefore, for all , there exist , such that, , having

For the above inequality, with a Cantor diagonalization argument, using Lemma 11(ii), , we know that there exists satisfying , such that going if necessary to exist a subsequence of satisfying

Then by the interpolation theorem, we obtain that

And by the weak low semicontinuity of the norm in , we also obtain that

and is a minimizer of problem .
Using an argument by contradiction, we prove

Assume that there exist and a subsequence of satisfying

By the above proof, we know that there exist a subsequence of , , and satisfying

So we derive

where since , we have that

Then there exists a contradiction. We completed the proof of Theorem 9.

Proof of Theorem 10. Now using an argument by contradiction, we complete the proof of Theorem 10. Assume that the set is not -stable. Then there exist , , and such that

and for satisfying (1) with initial value , we have

By (48), we know that

Then there exist satisfying and such that is a minimizing sequence of the problem . Since , and , we easily derive that is a minimizing sequence of the problem . By Theorem 9, there exists , for sufficiently large satisfying

Therefore by Lemma 11(ii), (49), , , and the above inequality, for sufficiently large , we derive that

Then obtaining a contradiction, the set is -stable with respect to (1). The proof is completed.

Conflict of Interests

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

Acknowledgments

This work was supported by the NSF of China (nos. 11371183, 11271050) and the NSF of Shandong Province (no. ZR2013AM004).

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