The Application Of Generalized Symmetry And Generalized Simplest Equation Method In Several Nonlinear Evolution Equations

Since the 1960s,nonlinear science is widely applied in natural sciences such as mathematics,chemistry,physics and other fields.At the same time a large number of nonlinear partial differential equation(PDEs)were appeared in people's horizons,and studying these nonlinear PDEs has become the primary task of developing nonlinear science.There are many directions of great theoretical and practical research on non-linear PDEs,so the study of nonlinear PDEs has important theoretical and practical significance.Because the complexity of nonlinear equation itself,the solution has a certain degree of difficulty.This paper will study the generalized symmetry(such as nonclassical symmetry,potential symmetry)and generalized simplest equation method in some nonlinear PDEs equation.The specific contents are as follows:The first chapter,we introduced the development of symmetry,generalized sym-metry(nonclassical symmetry,potential symmetry),determination of optimal system,Wu-differential characteristic set algorithm and generalized simplest equation method.The second chapter,we mainly studied the classical and nonclassical symmetric classification of a type composite equation.A new equations was composed that was a kind of complex equation with appending the invariant surface condition.And according to the Lie algorithm,the classical and nonclassical symmetric classification of the equation is determined,and the classical symmetry of the composite equation is expanded.In the classification process,8 kinds of F(u)that allows nonclassical symmetry were found,the results which obtained has enriched the symmetry and exact solutions of the equation.The third chapter,we mainly studied the optimal system and invariant solutions for two nonlinear evolution equations.First,the one-dimensional optimal system of the Poisson equation was calculated by the method which was proposed by P.J.Olver and the invariant solution of the Poisson equation was determined.Second,the classical symmetry was expanded by determining the potential symmetry of the coupled Burg-ers equation,the commutator title and adjoint title were calculated,one-dimensional optimal system of Coupled Burgers equation was obtained and the method for one?dimensional optimal system is proposed by Ovsiannikov.Combining with the structure of one-dimensional optimal system and invariant solution were constructed.These so-lutions can not be obtained from the classical symmetries of Poisson equation and Coupled Burgers equations.The fourth chapter,we mainly studied the application of the generalized simplest equation method.All kinds of exact solution of the Burgers equation was constructed by this method,these solutions include the traveling wave solutions,non-traveling wave solutions,multi-soliton solutions,rational solutions and other types of solutions.The traveling wave solutions and non-travelling wave solutions were mainly studied in this chapter.Finally,we summarize all the content of this paper,and look forward to the relevant research on the next step.