Application Of Computerized Symbolic Computation To Study On Solutions Of Several Nonlinear Models

Nonlinear phenomena has widely existed among the various areas of Natu-ral Science and Engineering Technology, and nonlinear science has been ap-plied in many fields, such as hydromechanics, optical communication, etc.Therefore, research on the nonlinear theory have been one of the hotspots in academia. The soliton theory, as an important branch of nonlinear science, has also attracted much attention of scientists. As for analytic study on the non-linear models, getting soliton solutions of related equations is a crucial step.In this paper, soliton solutions of some nonlinear models have been studied by Hirota bilinear method, Bell-polynomial method and B(?)cklund transformation method, and some other analytical properties, such as Lax pair and Infinite con-servation laws, have been studied. Moreover, computer symbolic computation is an important tool for getting solutions of nonlinear models.This dissertation can be divided into the following five parts:The first chapter is the introduction of the research background and status,including the development history and current situation of soliton theory. The basic knowledge of symbolic computation also has been introduced.The second chapter introduce the method we used in this paper to study the analytical properties of nonlinear evolution equations. They are Hirota bilinear method, Bell-polynomial method and B(?)cklund transformation method. The relevant theoretical principle and concrete steps are also studied.The third chapter is the research on a two mode KdV equation. First, we in-troduce an auxiliary variable, then via Hirota bilinear method, Bell-polynomial method and B(?)cklund transformation approach, its bilinear form, B(?)cklund transformation and N soliton solution are obtained. Through graphical anal-ysis, the features of soliton's propagation and collision are studied, and come to the conclusion that elastic collisions have been occurred between the solitons of TKdV equation.The fourth chapter investigates a variable-coefficient modified Kadomtsev-Petviashvili (mKP) equation, by introducing an auxiliary function, via Bell-polynomial method, Hirota bilinear method, and symbolical computation, its multi-soliton solution and B(?)cklund transformation are derived. Through the software "Mathematica",the shock, elevation solitary, and depression solitary waves described by multi-soliton solutions are analysed graphically. Soliton propagation and parametric conditions for the existence of the three waves are illustrated, the influence of variable coefficients are also discussed. At last, the elastic and inelastic collisions properties are revealed.The fifth chapter introduce the theoretical background and research sig-nificance of Lax pair and Infinite conservation laws. We choose 3+1 dimen-sional Jimbo-Miwa equation as study subject, propose its Bell-polynomial-type B(?)cklund transformation, and based on this BT, we present the Lax system and Infinite conservation laws of 3+1 dimensional JM equation.The sixth chapter is a summary of whole dissertation. We sum up our research work in this paper, and make the prospect of our future work and the problem we has met during the research.