New Exact Solutions for the Coupled Nonlinear Schrödinger Equations with Variable Coefficients

In this paper, coupled nonlinear Schrödinger equations with variable coefficients are studied, which can be used to describe the interaction among the modes in nonlinear optics and Bose-Einstein condensation. Some novel bright-dark solitons and dark-dark solitons are obtained by modified SineGordon equation method. Moreover, some figures are provided to illustrate how the soliton solutions propagation is determined by the different values of the variable group velocity dispersion terms, which can be used to model various phenomena.


Introduction
In nonlinear optics, the coupled nonlinear Schrödinger (CNLS) equations are often used to describe propagation of optical soliton in birefringence fibers, multimode fibers and optical fiber arrays. Many researchers have studied the CNLS equation with constant coefficient. In recent years, a number of methods are used to solve the coupled integrable nonlinear models, such as Hirota bilinear method [1] [2] [3], Painlev analysis method [4], Function expansion method [5] and direct perturbation method [6] and so on. However, the evolutions of vector solitons for CNLS equation with constant coefficients are not dependent on any controllable parameters. With the development of modern science, people need to manage and control soliton propagation, which will make solitons into practical information carriers. Therefore, variable coefficient equation has more prac-How to cite this paper: Qiu, Y.T. and Gao, P. (2020) New Exact Solutions for the Coupled Nonlinear Schrödinger Equations with Variable Coefficients. Journal of Applied Mathematics and Physics, 8, 1515-1523. tical significance.
In this paper, we will consider the following coupled nonlinear Schrödinger equation with variable coefficients (VCNLS) [ where 1 ψ and 2 ψ are complex envelopes of the propagating beam of the two modes, and x, t are the spatial coordinate and retarded time respectively. The coefficients ( ) a t represent the group velocity dispersion;  [9] have been a subject of great interest to mathematicians and physicists. Han [7] constructed an explicit transformation, which maps VCNLS to the classical CNLS, and obtains Bright-Dark solitons for VCNLS. Exact traveling wave and soliton solutions of the VCNLS equation have been obtained by Zhong [10] using homogeneous balance principle and the F-expansion technique. Yu and Yang [11] presented the similarity transformations for this system. Because of the complexity of VCNLS form, it is difficult to solve it directly, so the study of this kind of soliton solutions is not so extensive.
The paper is organized as follows. Section 2 describes the modified Sine-Gordon equation method [12]. In Section 3, we have applied the method to the VCNLS, and derive some bright-dark solitons and dark-dark solitons. Section 4 is devoted to analysis of shape changing exhibited by these soliton solutions when variable coefficients are altered. In the last section, Section 5, conclusion is presented.

The Method
Let us consider a form of a nonlinear partial differential equation . In the following, we offer the main steps of this method: Step 1: Use the following assumptions: where ( ) is the new independent variable, ( ) λ t is an arbitrary function of t and µ and α are the frequency and the width of the soliton respectively.
Step 2: Collect the coeffients of ( ) 1 ζ Journal of Applied Mathematics and Physics coefficient.
Step 4: The derivatives and powers of ( ) are equal to the term multiplied by a constant, so the arbitrary functions will be determined, and the Equation (2) is transformed into the following nonlinear ordinary differential system.
Step 5: Use the solutions of the Sine-Gordon equation [13] [14] by assuming that ( ) A and 0 D are arbitrary constants and n and m are determined by balancing the most dispersive term and the greatest nonlinear term in Equation (4), and Step 6: Equating the coefficients of obtained, by solving them with a Maple program and back-substituting into Equation (5) and Equation (3) via Equation (7), novel soliton solutions are obtained for the system of Equation (2).
Advantages of the method: The Sine-Gordon equation method has limitations and is suitable for some constant coefficient systems, but modified Sine-Gordon equation method is applicable to systems with variable coefficients containing imaginary parts. As a result, some spanking new solutions might be originated via this method and this method can use computational software like Maple or Mathematica to reduce the amount of computation.

Exact Solutions for VCNLS
By substituting the assumptions in Equation (3) into Equation (1), we obtain to make Equation (8) (13) where 1 c , 2 c , 3 c and 4 c are constants and 0 λ and 0 θ is an integration constant. Therefore, And, Equation (8) can be simplified as follows By balancing the dispersive and nonlinear terms in Equation (15) Substituting Equation (16) and the necessary derivatives into Equation (15) using Equation (6) constants: Solving Equations (17)-(25), we obtain the following cases and solutions using Equation (7).
Case 1: When 0 , we get the following bright-dark solitons: where 1 A , 1 D , 1 c and 2 c are arbitrary constants.
where 1 B , 1 D , and 1 c are arbitrary constants.
where 1 A , 1 E , 1 c and 4 c are arbitrary constants.

Physical Application
In this segment, we will illustrate the figure and designate the acquired solutions to the VCNLS equations. The solutions (26)-(29) come in terms of hyperbolic function. Next, we study the evolution behavior of the dark-bright soliton solutions given by Equation (26), the bright-dark soliton solutions given by Equation (28), and interaction of the two solutions given by Equation (28), illustrated in the figures.
In Figure 1, we shows the soliton solutions evolution of Equation (26) with different variable coefficients 1, t, t 2 , sin(t). Figure 1 In Figure 2, we shows the soliton solution evolutionis of Equation (28)  In Figure 3, we can obtain the similar results. Figure 3 demonstrates that   each soliton shape keeps invariant after interaction, which denotes that the interaction is elasticwe. We can see that the solitons show a periodic property but the solitons are not symmetrical in the t direction, and have the bell-shaped, parabolic, cubic or periodical-oscillating shapes.

Conclusion
In this paper, we have obtained some bright-dark soliton solutions and dark-dark  Figure 1 and Figure 2, and found that the propagation of the soliton solutions is determined by this value. We also investigated the evolution and interaction between the two solutions, and obtained that each solution shape keeps invariant after interaction and a periodic property in the t direction, as presented in Figure 3. The results show that modified sine-Gordon method gives soliton solutions for variable coefficients systems directly, without difficult calculations and also could be applied to many coupled nonlinear models.