Discussion Of The Exact Solution Of Nonlinear Evolution Equations For Certain Types Of Variable Coefficients

Under the guidance of Professor Sun Fuwei, this paper explore the exact solutions of nonlinear evolution equations based on the theory of soliton, It mainly studied a wide range of low-dimensional variable coefficient nonlinear equations and high-dimensional variable Exact solutions of linear coupled equations especially exact solitary wave solutions in many natural sciences applications.The first chapter introduces the domestic and foreign history and development of non-linear science, soliton theory focusing on brief exact solutions of nonlinear evolution equations in several major ways and relate methods of research status at home and abroad.The first part of the second chapter details the Painleve analysis of development at home and abroad and specific steps of using it gain exact solutions of nonlinear evolution equations. By the Painleve analysis method, the second part study variable coefficient nonlinear equations which is one frontier of the research of nonlinear evolution equations. For variable coefficient KdV-Burgers equation as an example, the paper get the variable coefficient KdV-Burgers equation and Backlund transformation expression and describe the nonlinear evolution equations by the exact solitary wave solutions determined by variable coefficients through simple image.In the basis of understanding research results in chapter II, the third chapter continues to explore the application of nonlinear evolution equations in high-tech areas:variable coefficient, high dimension, coupled system. With the Painleve analysis method, the paper has again studied high-dimensional nonlinear coupled coefficient equation. For 2+1 dimensional variable coefficient nonlinear Schodinger system as typical of the variable coefficient problems, it gains the exact solutions of expression and the Backlund transformation of high-dimensional coupled system and makes analysis of the exact solution that determined by this high-dimensional variable coupled equations, In the same way of the second chapter, it explains exact solitary wave solutions with images, which deepen with the Painleve analysis method to solve exact solutions of high-dimensional, variable coefficients, nonlinear coupled systems research..The forth chapter summarizes the research conclusions pointed out the complex issues of physical theory causes that explain accordingly adopted Painleve analysis approach with variable coefficient, high dimension, nonlinear evolution coupled equations in chapterⅡ, ChapterⅢ.The paper identifies the modifications and improvements in the above analysis and the physical meaning of exact solitary wave solutions of complex problems, extending the exact solutions of nonlinear evolution equations on the theory of solitons, At last, it mines independent innovation and weaknesses during the research and looks forward to the prospects for the development of nonlinear science.