Modified Exp-function And (G'/G)-expansion Method In The Application Of The Nonlinear Partial Differential Equations

In nonlinear systems,the soliton theory is an important research content.However,looking for solutions of the nonlinear equations is the main content in the soliton theory.To solve nonlinear partial differential equations is more difficult than to solve linear equations or ordinary differential equations.Although it has put forward many methods for solving nonlinear partial differential equations,they are not universal.Therefore,the study on solutions of the nonlinear equations is still of great significance.The main research contents of this paper are as follows:1.First,we introduce the existing method of solving nonlinear partial differential equations.Then we introduce the main steps of the homogeneous balance method,(G'/G)-expansion method and Exp-function method to solve the nonlinear partial differential equations.We also give various solutions of the Riccati equation ?'=A + B?+C?2.2.According to the(G'/G)-expansion method,we acquire the modified(G'/G)-expansion method.The new form of solutions can be expressed as follows:where G = G(?)satisfies Riccati equation.Then,two equations are solved by the modified(G'/G)-expans ion method.One is mBBM equation ut +ux-3?u2ux-uxxt= 0,the other is PC equation utt-uttxx-(ru2n+1+pun+1 +qu)xx=0(n=1).We get a lot of solutions,and some pictures of the solutions are presented with reasonable assignment.3.According to the Exp-function method,we acquire the modified Exp-function method.The new form of solutions can be expressed as follows:where F = F(?)satisfies Riccati equation.Then,three equations are solved by the modified Exp-function method.One.is MEW equation ut + p(u3)x-quxxt =0;the second one is ZKBBM equation ut+ux-2puux-quxxt=0;the third one is Boussinesq equation utt-?2uxx-?(u2)xx+uxxxx=0.We get rich solutions,and some pictures of the solutions are presented with reasonable assignment.