Function Expansion Method For Solving Nonlinear Partial Differential Equations

Due to the rapid development of science and technology, many problems in physics can be summed up in mathematical model. As to the nonlinear partial differential equations of the mathematical model, searching for their exact solutions not only has a very important position in the theory of solitons, but also help us to deepen our understanding of the essence of physics. In solving nonlinear partial differential equation, function expansion method has an irreplaceable role, among them, the generalized tanh function expansion method and consistent Riccati expansion method are quite effective. In this thesis, we will search the exact solutions of the Gardner equation,(2+1)-dimensional Konopelchenko-Dubrovsky equation, a Boussinesq-type coupled equation by applying these two approaches.In the study of the Gardner equation, the solution and the consistent equation of the Gardner equation are obtained by the generalized tanh function expansion method. Solving the consistent equation by these two approaches, we can obtain the soliton-cnoidal wave interaction solutions.As to the Gardner equation, ?rstly we will introduce a new approach called consistent Riccati expansion, a new concept which can be obtained from the consistent Riccati expansion called consistent Riccati expansion solvability. The consistent equation can be given by using the consistent Riccati expansion approach. A kind of exact solution with special structure of the the Gardener equation can be obtained.In the study of the(2+1)-dimensional Konopelchenko-Dubrovsky equation, with the help of the consistent Riccati expansion method, the consistent equation and the solution of the(2+1)-dimensional Konopelchenko-Dubrovsky equation can be obtained, eventually, we found the interation sloution are period-kink by solving the consistent equation.In the study of the a Boussinesq-type coupled equation, the consistent equation and the solution of a Boussinesq-type coupled equation can be obtained by applying the consistent Riccati expansion method, then the solution among soliton and other types of nonlinear excitations of a Boussinesq-type coupled equation can be obtained, it is noteworthy that by means of the Jacobi elliptic functions and the third type of incomplete elliptic integrals, the soliton-cnoidal wave interaction solutions are periodic-kink.