Application Of The Function Expansion Method In Solving Nonlinear Partial Differential Equations

Finding solutions to nonlinear partial differential equations is an ancient and important research topic.In recent years,through the efforts of many mathematicians and physicists at home and abroad,many methods for solving nonlinear partial differential equations have been proposed.For example,the hirota bilinear method,the function expansion method,the homogeneous balance method,the first integral method,the function transform method,the polynomial heuristic method and so on.We have noticed that the polynomial heuristic method,homogeneous equilibrium method and function transformation method can be regarded as the improved form of function expansion method.In this paper,the exact solutions of three kinds of nonlinear partial differential equations are studied by using three improved forms of the function expansion method.In chapter 1,We recall the research history of nonlinear partial differential equations,and several kinds of the function expansion methods for exact solutions are summarized.We state our main results of three kinds of nonlinear partial differential equations.In chapter 2,we study the nonlinear reaction diffusion equations,it can serve as the general form of the Huxley equation,the Chaffee-Infanfe equation,the Fitzhugh-Nagumo equation and the generalized Fisher equation.We used the method of polynomial heuristic to get six new type tanh travelling wave solutions and a series of new type coth travelling wave solutions.Then,we use the result.Finally,we obtained the Fitzhugh-Nagumo four new travelling wave solutions of equation and the generalized Fisher two new travelling wave solutions of the equation.In chapter 3,we study the(2+1)dimensional Jaulent-Miodek equation.It involves many branches of physics,such as plasma physics,fluid dynamics and so on.We use the homogeneous balance method to get the four new solitary wave solution of the equation.In particular,we obtained two new kink soliton solutions and two new singular kink soliton solutions.In chapter 4,we study the higher-order nonlinear Schr?dinger equation,which can be used to describe the propagation of femtosecond soliton pulse in optical fiber communication.We use the function transformation method to obtain many explicit solutions of the equation.