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

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 thatSo 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 satisfyApparently, there exists , satisfying where because of definition of and (27).
Setting , they satisfy that ; ; for , ; and . Then we have thatTherefore (29) imply that there exist and , , satisfying , . Then we have thatUsing Lemma 7, we derive thatwhere . By , the Sobolev inequality, Lemma 11, and (31), we obtain that for sufficiently large . Similarly, for sufficiently large we have that So we have thatBy Lemma 11, (16), (27), , and the definitions of , (), we can derive that where and . Therefore by the two inequations, we obtain thatBy (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 thatand for satisfying (1) with initial value , we haveBy (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).