Computer Applications to Exact solutions of AKNS Hierarchy with Variable Coefficients

Abstract

It is well known that computer has been widely used in almost all fields. In this paper, Hirota’s bilinear method with computer symbolic computation is extended to a new AKNS hierarchy with variable coefficients. As a result, one-soliton solutions and two-soliton solutions are obtained. Through the analysis of the obtained one-soliton solutions and two-soliton solutions, a uniform formulae of n-soliton solutions are derived. It is shown from computer running that the obtained one-soliton solutions and two-soliton solutions satisfy the AKNS hierarchy. Introduction In the past several decades, some powerful symbolic computation systems like Mathematica or Maple have become indispensable auxiliary tools for solving nonlinear partial differential equations (PDEs) [1-8]. Finding exact soliton solutions of nonlinear PDEs plays an important role in the study of nonlinear physical phenomena and has gradually developed into a significant direction in nonlinear science. Some effective methods for constructing multi-soliton solutions have been proposed, such as those in [9-11]. Among them, Hirota’s bilinear method [11] is a purely algebraic method, the process of which is fairly simple and convenient for computer operation. Since the Hirota’s bilinear method was proposed, the method has received extensively applications [12-15]. However there are very few studies on generalizing the method for a whole hierarchy of nonlinear PDEs. In this article, we will extend the Hirota’s bilinear method to a new AKNS hierarchy with variable coefficients