Research Of Bifurcation Theory And Traveling Wave Solution Of Several Nonlinear Evolution Equation

In this paper,the bifurcation theory of dynamical systems and Hirota bilinear method are used to study some nonlinear evolution equation.Firstly,the phase protraits of a nonlinear dispersion equation is obtained by using the bifurcation theory of dynamical systems.Secondly,a nonlinear partial differential equation is derived from a plane evolution curve,with the use of Lie group analysis,its one-dimensional optimal system is obtained.Besides,the phase protraits and exact solutions are also given by the method of plane dynamical system.Finally,Hirota bilinear method is used to study a KP equation and kinds of exact solutions are expressed.In the firt part,the extend nonlinear dispersion m K(m,n)equation is transformed into a nonlinear ordinary differential equation by introducing a travelling wave transformation and it can be rewritten into an ordinary differential system.The phase protraits of the ordinary differential system is given by using the planar dynamical system method,and the expression of the exact solution of the equation is given.And on this basis,the exact solutions are obtained.In the second part,a nonlinear partial differential equation is derived from a plane evolution curve and one-dimensional optimal system of the equation is given with the use of revelent content of Lie group.We also apply plane dynamicla system method to study this equation and its phase protraits and explicit exact solutions are also obtained.In the fainlly part,a KP equation is solved by using of Hirota bilibear method.We use bilinear operator to transform the KP equation into a bilinear form.And setting the solutions of equaiton as different form,different types of solutions(N-solition soluiton,breather wave solution,rogue wave solution).