Research On Solutions Of Some Higher Order Nonlinear Evolution Equations

In this thesis,the symmetry of a class(1+1)dimensional CDGSK equation is obtained by using Lie group method,and the optimal system of the equation is obtained according to its symmetry.The invariants and group invariant solutions of the equation are constructed by using the optimal system,and four kinds of reduced ordinary differential equations are obtained.In order to obtain the solution of the reduced equation,some new exact solutions are obtained by means of traveling wave transformation and function expansion,and the images of the solution are given.Secondly,the Hirota bilinear method is used to study the equation,the N-soliton solution of the equation is obtained,and the image of the 2-soliton solution is given.Thirdly,the existence of global solutions of Cahn-Hilliard equation is discussed,and the images with different initial values are obtained by using finite difference method.Finally according to the basic idea of Darboux transformation,a class of high order nonlinear coupled AKNS system is solved.The first chapter,the symmetry of(1+1)dimensional CDGSK equation is constructed by Lie group method,and the optimal system of(1+1)dimensional CDGSK equation is obtained according to obtained symmetry,and the invariants and group invariants of the equation are given.Then the(1+1)dimensional CDGSK equation is reduced,and the equation is transformed into nonlinear ordinary differential equation or linear ordinary differential equation.Some new exact solutions of the equation are obtained by Riccati function expansion method.In the second chapter,the bilinear method is used to transform the(1+1)dimensional CDGSK equation into bilinear equation,and the original bilinear equation is solved by analogy with perturbation method,and the N-soliton solution of the equation is obtained.Finally,the image of the 2-soliton solution of the equation is given and discussed.In the third chapter,the existence problem of global solution of Cahn-Hilliard equation is proved by the method of prior estimation and sobolve’s first and second embedding theorem,and several three-dimensional images with different initial values are given by finite difference method.In the forth chapter,a class of high-order non-local coupled AKNS system is derived according to AKNS system.Then,the Darboux transformation of this kind of high-order non-local coupled AKNS system is obtained by using gauge transformation.Finally,the exact solution of this kind of high-order non-local coupled AKNS system is obtained according to Darboux transformation and the graph is made.