Solving Several Kinds Of Nonlinear Evolution Equations By Extended Homoclinic Test Approach

The nonlinear evolution equation plays an important role in the study of the nonlinear phenomena and the methods to solve nonlinear evolution equations have been improving. In this work, the solitary wave solutions of the nonlinear evolution equations are obtained by using the extended homoclinic test approach. The dissertation is divided into the following five chapters.The first chapter is the introduction. The research status of .nonlinear evolution equations and the extended homoclinic test approach are introduced.A simple introduction of discrete nonlinear equations and the main work of this thesis are described.The second chapter is the application of the homoclinic test method to solve the equations with constant coefficients. Through constructing the test function and taking the constant equilibrium solution, the solitary wave solutions of the (2+1 )-dimensional dissipative Zabolotskaya-Khokhlov equation are obtained by using the extended homoclinic test approach.Two-solitary wave solutions are analyzed and new mechanical feature are found. Through constructing the test function, the exact general solutions of the Boussinesq equation were obtained by using the extended homoclinic test approach. Breather solitary wave solution and rational breather wave solution of the Boussinesq equation were constrcted by using the homoclinic breather limit method which also shows that the rational breather wave solution was just a rouge. wave solution of Boussinesq equation.The third chapter is the application of the humoclinic test method to solve the nonlinear equations with variable coefficients. The (2+1)-dimensional variable-coefficient Zakharov-Kuznetsov equation and BKP equation are solved by the method. By introducing new test function we shall present a generalization form of the extended homoclinic test approach which was used to find several kinds of exact solutions for the (2+1)-dimensional variable-coefficients Zakharov-Kuznetsov equation and BKP equation from its Hirota bilinear form. These solutions involve the periodic soliton-like solutions, soliton-like solutions and periodic-like solutions.The fourth chapter is the application of the homoclinic test method to solve the nonlinear difference-differential equations and the fractional nonlinear partial differential equations. New test functions and Maple software are used to give the exact solutions of the discrete KdV equation and Toda lattice equation in the second section of this chapter. The fractional order partial differential equation is transformed into an integer order partial differential equation by using the suitable transform and then the periodic solitary wave solutions and two-solitary wave solutions of fractional order KdV equation are obtained by means of the extended homoclinic test approach.The last chapter is the summary of our work and the introduction of our future research work.