Analytical Study On Several Nonlinear Evolution Equations Based On Symbolic Computing

Nonlinear evolution equations are used to describe nonlinear phenomena in the fields of plasma physics,optical fiber communication,fluid mechanics and so on.Solving nonlinear evolution equations is of great significance in these fields.In the 1950s,researchers proposed the concept of "soliton" in the process of exploring nonlinear phenomena and studied the characteristics of "soliton".With the further study of soliton,many methods for finding exact solutions of nonlinear equations have been explored.Hirota bilinear method is one of the most classical and direct methods for finding exact solutions.A great deal of symbolic computation is involved in the process of solving nonlinear equations.The computational process has a certain repeatability and regularity.The emergence of computer algebra brings convenience to people's research work.The use of computers can not only improve the speed of computation,so that people can get rid of much repeated computation,but also check the computed results to ensure the correctness.Symbolic computation plays an important role in promoting soliton from theoretical research to practical application.The research work of this thesis includes the following five aspects:In chapter one,the history,development and research status of soliton are introduced.Symbolic computation and application of computer software in solving nonlinear equations are briefly introduced.In chapter two,we introduce several methods in the process of studying soliton problems,including Hirota bilinear method,Backlund transformation method and multiple exponential function method.Taking the classical KdV equation as an example,this chapter introduces the process of transforming the nonlinear equation into bilinear equation by using several related variable transformations.The process of constructing Backlund transformation of the KdV equation is also introduced.In chapter three,for the exponential traveling wave solutions of Hirota bilinear equation,a sufficient and necessary criterion for the existence of linear superposition principle has been given.Motivated by this criterion,we propose a new Hirota bilinear equation by using a multivariate polynomial.Applying the linear superposition principle to this new Hirota bilinear equation,we finally find two types of resonant multiple wave solutions,among which,the resonant three-wave solutions are illustrated with three-dimensional plots.In chapter four,two(3+1)-dimensional nonlinear evolution equations are studied analytically.The following work has been done for the first equation:(1)With multiple exponential function method and symbolic computation,non-resonant multiple wave solutions of the equation are obtained;(2)A bilinear Backlund transformation of the equation is constructed,the exponential function solution and the first order polynomial solution of the equation are obtained by using the Backlund transformation;(3)Considering the reduction of dimension corresponding to y=x and y=z respectively,the lump solutions of two reduction cases are studied;(4)The interaction between lump wave and strip wave is studied,and the characteristics of the solution are analyzed from mathematical expressions and graphics.For the second equation,the following work has been done:(1)A bilinear Backlund transformation of the equation is constructed,the exponential function solution and the first order polynomial solution of the equation are obtained;(2)Considering the reduction of dimension corresponding to z=x and z=y respectively,the lump solutions of two reduction cases are studied;(3)The interaction between lump wave and strip wave is studied,and the characteristics of the solutions are analyzed from mathematical expressions and graphics.In chapter five,a summary of the work in this thesis is given.Some difficulties encountered in the study of soliton are presented.In view of these difficulties,the future work is prospected.