Study On Solutions Of Nonlinear Evolution Equations And Their Applications Based On Symbolic Computation

Nonlinear science, a great development in natural sciences after quantum mechanics and the relative theory, is applied widely in mathematics, physics and other fields. In particular, the soliton theory, which is one of the three major branches of the nonlinear theory, has attracted the attention of many scientists and has been one of the hotpots of the academia. A major focus of research on the soliton theory is how to solve the nonlinear evolution equations to get soliton solutions. With further development of the soliton theory, scientists have made new breakthroughs in solving the exact solutions of nonlinear evolution equations, proposing various methods of constructing the exact solutions of the equations, such as the Hirota bilinear method, inverse scattering, Backlund transformation, Bell-polynomial approach and Wronskian determinant.The essay, based on the Hirota bilinear method, Bell-polynomial approach and Backlund transformation, studies the interaction properties between the blinear forms, Backlund transformations, and the solitons of the (3+1) dimentional B-type Kadomtsev-Petviashvili (BKP) equation and the (3+1) dimentional coupled nonlinear Schrodinger equation through symbolic computation.The chapters and main contents of this paper are organized as follows:The first chapter introduces the history of the soliton theory, its current development situation, basic properties, symbolic computation approach and the common methods to solve the nonlinear evolution equations, such as inverse scattering, Painleve analysis, and Wronskian technique.The second chapter mainly discusses the bilinear method, including the definition and properties of bilinear operators, three common forms of transformation in the bilinear theory, namely, rational transformation, logarithmic transformation and double log transformation, and how to get the soliton solutions of equations through the bilinear approach.The third chapter introduces Bell-polynomial and Backlund transformation, including the definition of Bell-polynomial and Backlund transformation and ways of obtaining the Backlund transformation.The fourth chapter is the study on the (3+1) dimentional BKP equation. Via Bell-polynominial approach, its bilnear form and Backlund transformation are obtained. By employing the method of small parameter expansions, its single and double soliton solutions are obtained. Through graphical analysis of Mathematica software, the propagation and collision properties of solitons are revealed.The fifth chapter focuses on a (3+1) dimentional coupled nonlinear Schrodinger equation in optic fiber communications. Its physical meaning and background are discussed. Its bilinear form, single and double soliton solutions are obtained and illustrated by graphs. The elastic collision and inelastic collision between the solitons are analyzed.The final chapter is a summary of the main points in the paper.