Darboux Transformation And Exact Solutions Of Some Nonlinear Soliton Equations

At present,solving the nonlinear evolution equations is one of the important research contents in applied mathematics and mathematical physics,and it is widely used in dynamics,superconductivity,meteorology,nonlinear physics,quantum field theory,communication and other fields.There are many methods for solving the nonlinear soliton equations,however the choice of method is a very difficult problem.Based on the Darboux transformation,Hirota bilinear method and the symbolic computing system MATHEMATICA,this thesis studies several nonlinear partial differential equations.The main work is as follows:The first chapter,the background,current situation,significance and research methods of the theory of integrable system,soliton theory,Darboux transformation,Hirota bilinear method and B¨acklund transformation are briefly described.The second chapter,the nonlocal coupled nonlinear Schr(?)dinger equations are studied.By constructing its Darboux transformation,the exact solution and iterative formula of the equations are obtained.Finally,the image of the solutions are depicted by the symbolic computing system MATHEMATICA.The third chapter,the classical multicomponent Schr(?)dinger equations are studied.Firstly,the Darboux transformation of N-component local nonlinear Schr(?)dinger equations are constructed,the exact solution and iterative formula of the equations are obtained.The Darboux transformation of the local nonlinear Schr(?)dinger equations with three components are constructed,and the exact solution and iterative formula of the equations are obtained.Then,the image of the solutions are depicted by the symbolic computing system MATHEMATICA.Secondly,the Darboux transformation of the nonlocal nonlinear Schr(?)dinger equations with three components are constructed,and the exact solution and iterative formula of the equations are obtained.Then,the image of the solutions are depicted by the symbolic computing system MATHEMATICA.The fourth chapter,the Drinfeld-Sokolov-Wilson(DSW)system was studied.Firstly,Hirota bilinear form of the system is obtained through the Bell polynomial and the exact solutions of the system are obtained,and used the symbolic computing system MATHEMATICA to draw the image of the solutions.Finally,the Lax equations of the system are obtained by Hirota bilinear form.The fifth chapter,the content of this paper is summarized,and the future work of further research is prospected.