Dynamical Behavior and the Classification of Single Traveling Wave Solutions for the Coupled Nonlinear Schrödinger Equations with Variable Coefficients

Abstract

In this paper, the dynamical properties and the classification of single traveling wave solutions of the coupled nonlinear Schrödinger equations with variable coefficients are investigated by utilizing the bifurcation theory and the complete discrimination system method. Firstly, coupled nonlinear Schrödinger equations with variable coefficients are transformed into coupled nonlinear ordinary differential equations by the traveling wave transformations. Then, phase portraits of coupled nonlinear Schrödinger equations with variable coefficients are plotted by selecting the suitable parameters. Furthermore, the traveling wave solutions of coupled nonlinear Schrödinger equations with variable coefficients which correspond to phase orbits are easily obtained by applying the method of planar dynamical systems, which can help us to further understand the propagation of the coupled nonlinear Schrödinger equations with variable coefficients in nonlinear optics. Finally, the periodic wave solutions, implicit analytical solutions, hyperbolic function solutions, and Jacobian elliptic function solutions of the coupled nonlinear Schrödinger equations with variable coefficients are constructed.

Superposition of modulated nonlinear waves in inhomogeneous systems with negative coherent coupling

“…As a very important physical model, CNLS equations with variable coefficients are employed to investigated the dynamics of the nonlinear waves in inhomogeneous fiber systems [25][26][27]. The solutions of the coherently CNLS equations with variable coefficients are difficult to obtained and usually derived under specific constraint conditions.…”

The exact solutions of Schrödinger-Hirota equation based on the extended auxiliary equation method

“…Nonlinear partial differential equations (NLPDE) [1,2] have always played a key role in scientific research. Many phenomenons and models in mathematics, physics, natural sciences, engineering, biology, social sciences, and computational sciences and other fields are described by PDE.…”

Classification of All Single Traveling Wave Solutions of ( 3 + 1 )-Dimensional Jimbo-Miwa Equation with Space-Time Fractional Derivative

“…Moreover, in order to make these NLPDE better fit the actual situation, many researchers also simulate complex factors by using mathematical tools, such as stochastic differential and fractional differential [1][2][3][4][5]. In the research methods, traveling wave solutions, as a special kind of analytical solutions of NLPDE, plays an important role in understanding nonlinear wave phenomena [6][7][8][9][10][11][12]. Therefore, the exact traveling wave solution is an attractive work in the study of theory and practice.…”