The Application Of Function Expansion Method In Two Nonlinear Equations

With the development of modern science and technology,the nonlinear physical equations play an important role in many fields of nonlinear science,such as plasma,fluid mechanics,medicine,biology and so on.Therefore,in order to combine mathematical with physics,the study of the solutions of non-linear physical equations have become one of the vital subjects of mathemat-ical researchers.At present,many methods of studying exact solutions have been proposed,for instance,the inverse scattering transforms method,Darboux transformation,the tanh-coth method,homogeneous balance method,bilinear method,exp-function method,sine-cosine method and so forth.These methods are used to solve the nonlinear physical equations,a large number of results are obtained,at the same time,a mass of new solutions are gained.In this paper,combined with Riccati equation,we research two kinds of non-linear partial differential equations and gain the rich wave solutions,by means of the tanh-coth method,with three different ansatz,that is radical sign combined ansatz,positive-power ansatz,positive and negative power-symmetric ansatz.In the beginning,we introduce consistent Riccati expansion and the tanh-coth method for the nonlinear physics equations.In the next place,the CRE inte-grability of Boiti-Leon-Pempinelli equation is proved by using consistent Riccati expansion.On the basis of tanh-coth method,we obtain some exact solutions and the interaction solution between its soliton and other excitation.In the end,the tanh-coth method is used to solve the Jimbo-Miwa equation,and the hyper-bolic function solutions,trigonometric function solution and rational function solution of this equation are obtained.