Exact Solutions And Approximate Solutions To Some Nonlinear Mathematical Physics Equations

Finding solutions to nonlinear mathematical physics equations is a very important problem in nonlinear science. Both the exact solutions and approximate solutions, their importance for practical problems lies in the reliable evalution of practical model in numerical simulation and the explanation of practical physical phenomenon. Therefore, it is studied widely. Many mathematicians and physicists work in the field, and they have developed many methods for many special equations. Particularly, by algebrac expansion method, many exact solutions of many nonlinear equations have been obtained. There are many methods for finding approximate solutions, too. For example, perturbation method , homotopy analysis method, and so on. In this paper,we study mainly these problems and solve exact solutions and approximate solutions of some nonlinear mathematical physics equations. First, using the method of complete discrimination system for polynomial, a number of exact traveling wave solutions to four nonlinear mathematical physics equations are obtained, which include Dodd-Bulloug-Mikhaillov equation, modified BBM equation, combined KdV-mKdV equation, Fuchssteiner-Focas-Camass-Holm equation. Some solutions were new. Second, using Liu's trial equation method, many single traveling wave solutions four nonlinear mathematical physics equations ,i.e., modified Kawachara equation, KK equation, combined KdV equation, Getmanou equation are given. Among those solutions, some are new. At last, using homotopy analysis method, we can set up homotopy which is a equation that contains a embedded parameter, then we obtain the second order approximate traveling wave solution to mKdV-Burgers equation, this solution is new.