New optical soliton solutions for the variable coefficients nonlinear Schrödinger equation

This paper is devoted to seek new optical soliton solutions of nonlinear Schrödinger equation (NLSE) with time-dependent coefficients which describes the dispersion decreasing fiber. To achieve optical soliton solutions of NLSE, the basic idea of homogenous balance approach has been used to propose Bernoulli (G′/G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(G'/G)$$\end{document}-expansion method, where G=G(ζ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G =G(\zeta )$$\end{document} satisfies Bernoulli equation, which is easier to solve than previous studies. By applying some transformations and using this method, some periodic wave, bright and dark soliton solutions are successfully obtained. Moreover, 3D surfaces, standard deviation line plots and contour maps graphs of the obtained results under effect of different values of coefficients are illustrated to have acceptable image of dynamic structures and to find the relation between the parameters and wave behaviors.


Introduction
The variable coefficients nonlinear Schrödinger equation (VcNLSE) is one of the main topic in modern physics to pragmatically describe many nonlinear behaviors in nonlinear optical fibers, Bose-Einstein condensation, plasma and water waves (Ponomarenko and Agrawal 2007;He et al. 2009;Yu and Yan 2014;Yan et al. 2015;Yao et al. 2016;Kaur and Wazwaz 2019). Several studies have been done to peruse on solutions of VcN-LSE; for example, Pérez-García et al. (2006) used similarity transformations connecting NLSE involved time-varying coefficients with the autonomous cubic NLSE. He et al. (2014) studied rogue wave solutions in NLSE with variable coefficients. Kedziora et al. (2015) by using Lax pair and Darboux transformation formalisms could find the solutions of VcNLSE. Liu et al. (2019) achieved the distinguished types of nonautonomous complex wave solutions, which include bright and dark soliton solutions. Guo and Liu (2020)) by analyzing the dynamical properties of the related Hamiltonian obtained some new dark solitons, bubble solutions and periodic solutions. El-Shiekh (2019) modified direct similarity reduction method to find Jacobi, hyperbolic and periodic wave solutions of VcNLSE.
In the past few years studying on different types of VcNLSE with coefficients which depend on the evolution variable has been of great interest to scientists. For instance, stabilized solitons appear in multidimensional VcNLSE equations when the coefficient is controlled appropriately, that exists in optical applications (Bergé et al. 2000), according to these projects, it was realized that there is a similar phenomenon arisen in the context of mean field models of Bose-Einstein condensation (Abdullaev et al. 2003;Saito and Ueda 2003;Montesinos 2004). The studies of the nonlinear waveforms for the nonlinear Schrödinger equations with external potentials are of great significance the rapid development of the Bose-Einstein condensates (Wazwaz 2009;Bao and Cai 2012;Feng 2016;Feng and Zhang 2018). For the nonlinear Schrödinger equations with external potential, Zhang (2000) studied the existence of the condensations in a critical value. Bao and Cai (2012) accomplished numerical studies on the nonlinear Schrödinger equations with singular potential modelling the dipolar Bose-Einstein condensation for ground states.
In this research, we explain the optical soliton solutions of the following form VcNLSE with time-dependent coefficients (Serkin and Hasegawa 2000;Kruglov et al. 2005;Zeng et al. 2016) by proposing Bernoulli (G � ∕G)-expansion method: here (x, t) is a complex wave function of x and t, the dispersion parameter (t) , the nonlinearity parameter g(t) and gain/loss parameter (t) (> 0, < 0) are real functions of time. One of physical significance of Eq. (1) is that it is used as a model for the Bose-Einstein condensate dynamics, which is considered by the mean field approximation when the nonlinear coefficient g(t) is controlled by applying Feshbach resonances. The proposed method of this paper is inspired by the (G � ∕G)-expansion and the generalized (G � ∕G)-expansion methods Zhang et al. 2008), which is based on the ideas that traveling wave solutions of NLEE can be explained by a polynomial in (G � ∕G) , and G = G( ) satises a second order linear ordinary differential equation with constant coefficients, while in Bernoulli (G � ∕G)-expansion method, we express G = G( ) as a Bernoulli equation with variable coefficients. Although many useful methods are developed to investigate NLEEs, our method enrich the studies of this area and the results show that the proposed method is effectual to search exact solutions of many other nonlinear differential equations with variable coefficients. (1) 2 Description of the Bernoulli (G � ∕G)-expansion method A NLEE with dependent and independent variables, respectively u and X = (x, t) , is given by By using transformation, Eq.
(2) can be reduced to an ordinary differential equation (ODE), which the solutions can be expressed by where a 0 (X), a i (X) (i = 1, 2, … , n) should be determined, and = (X) which in this paper we conider it as = b (x − c t), where b and c are constants. Function G = G( ) satisfies the following Bernoulli equation: where p( ) and q( ) should be specified. By balancing the highest order nonlinear term and the highest order derivative of u in ODE, integer n will be achieved. Substituting (3) along with Eq. (4) into ODE and collecting the same order of G together will get a set of algebra equations of variable coefficients that can be determined by use of Maple, while general solutions of (G � ∕G) are given as where C 1 is an arbitrary constant. Substituting these results into (3), we have exact solutions of Eq. (1).

Applications
The first step to find exact solutions of Eq. (1) is done by applying the following transformation where is amplitude function, k and are phase constants. By substituting (6) into (1), for real and imaginary parts respectively, we have Balancing between xx and 3 in (7), can get n = 1 for (3), which leads to as follows: where G = G( ) satisfies Eq. (4). In this research, we have studied on two different conditions to derive exact solutions of Eq. (1): First condition: Substituting (10) into (7) and (9), collecting coefficients with the same order of G j , (j = 0, 1, 2, 3) to zero, yields a system of differential equations as follows: which have solutions where C 2 , C 3 and C 4 are arbitrary constants. (t) is an arbitrary function. In this study, it is considered as follows with 0 and as constants.
Substituting the above results into (10) and by use of (6) and (5), we have: where is defined by b (x − c t). Second condition: Considering b = 1, c = 0 and k = 0, and substituting (10) into (7) and (9), collecting coefficients with the same order of G j , (j = 0, 1, 2, 3) to zero, yields a system of differential equations as follows: Solving this system of differential equations, gives us two different cases: Case 1.
where p, q, and C 2 are arbitrary constants. Substituting the above results into (10) and by use of (6) and (5), for wave function, we have: 255 Page 6 of 12 where C 2 , C 3 , C 4 , C 5 , C 6 and are arbitrary constants. By setting these results into (10) and by use of (6) and (5), wave function can be obtained as:

Dynamic structures
In this section we will demonstrate the dynamic structures of our results to get better understanding of wave solutions. Figure 1 illustrates the wave propagation of wave function 1 (x, t), by using = 1.
3 (x, t) = − 1 2 e ∫ 0 e t dt e −i t . and c = −2.5 with three different values of 0 and as follows: Fig. 1 (a1-a3) describe 3D, line and contour maps images for 0 = −0.2, = 0.11. Fig. 1 (b1-b3) describe 3D, line and contour maps images for 0 = 0, = 0, which show us the results when (t) = 0. We also represent in Fig. 1 (c1-c3), 3D, line and contour maps images when 0 = 0.2, = 0.11. In these figures, we can see periodic behaviors of the wave that is by considering negative or positive values of 0 respectively, the height of wave decreases or increases over time. Figures 2 and 3 are displayed to show the important role of constant c in dynamic behaviors of wave function 1 (x, t), respectively by seting c = 0.1 and c = 2.5 while other parameters are same as Fig. 1. In Fig. 2 for c = 0.1 we have solitary waves that include several peaks, which for different values of 0 , they show decreasing or increasing behaviors across the time. In Fig. 3 by setting c = 2.5, compared to Fig. 1, we have a different kind of periodic wave plots with milder waves and these waves have diagonal motion across the x and t coordinates. Figure 4 exposes the solution of 2 (x, t) in Eq. (1) by using = 1.2, p = 2, q = 1.1, C 1 = 0.01 and C 2 = 1.41 with three different values of 0 and as follows: Fig. 4 (a1-a3) describe 3D, line and contour maps graphs for 0 = −0.2, = 0.11. Fig. 4 (b1-b3) describe 3D, line and contour maps graphs for 0 = 0, = 0, which show us the results when (t) = 0. We also show in Fig. 4 (c1-c3), 3D, line and contour maps graphs by setting 0 = 0.2, = 0.11. As we can see, these figures demonstrate dark soliton solutions with different heights in different 0 values.

Conclusion
In this article, we have presented Bernoulli (G � ∕G)-expansion method to find optical soliton solutions of VcNLSE, based on expressing exact solutions of NLEE by a polynomial, which is defined as a fractional function of the Bernoulli equation with p and q as variable coefficients. Due to the complexity of the computation, by use of Maple we could get the explicit output of solutions. To our knowledge, the begotten results have not been reported in another literatures. This method could easily explain exact solutions of VcN-LSE in two conditions, and also it can be applied to the other NLEEs with space-dependent or time-and space-dependent coefficients.
In addition, to complete our research, we have demonstrated the dynamic structures of the obtained wave functions by providing an arbitrary function with different values of the free parameters. We could also show some periodic behaivor of waves, bright and dark soliton solutions and utilize MATLAB to help us to clearly depict the curvature of the waves. These illustrations showed us the relation between parameters and waves behaviors, in which, considering the negative values of 0 , the wave functions decrease over time and for positive values of 0 , the wave functions icrease with time.