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

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.


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][2][3][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.
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][10][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][13][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][16][17][18][19][20][21][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 2-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.

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 ( 1 (, ),  2 (, )) = (  1   1 (),   2   2 ()), where   ∈ R ( = 1, 2).Then we only need to find  1 ,  2 ,  1 (),  2 () satisfying the following equations to study the existence and stability of standing waves of (1): where  1 ,  2 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   (R  ) 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 . Throughout this paper, we denote the norm of 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 0 <  < 1,  ∈ R  , and  ∈ ,  representing the Schwartz class, then the fractional Laplacian (−Δ)  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 0 <  < 1, there are two properties with   (R  ): , where  ,2 (R  ) is defined by the method of trace interpolation (see chapter 7 of [24]) and its norm is given by ‖‖  ,2 (R  ) . and  are positive constants.
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 0 <  < 1, ,  ∈ , the Schwartz class, then where 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 2  → +∞, we can obtain the following theorem.