Expansion Method About The Jacobi Elliptic Function And Its Applications To Solving Nonlinear Partial Differential Equations

Nonlinear partial differential equation is different from the linear differential equation, and it is impossible to have unified method of solving nonlinear partial differential equation. In order to obtain exact periodic solutions of nonlinear partial differential equation, many kinds of effective methods are put forward, for example, Hirota bilinear operator, Backlund transformation method, Darboux transformation, Painleve truncated expansion method, variable separation approach, similarity reduction, algebraic geometry method, inverse scattering method, Cole-Hopf transformation and function expansion methods and so on. Especially, people pay more and more attention to using Jacobi elliptic function expansion method for solving the exact solution of nonlinear partial differential equation. The main content of this paper is as follows:1. The paper summarizes the concepts and property of Jacobi elliptic function, the property of Lame equation and Lame function. It also introduces the main steps about Jacobi elliptic function expansion method.2. The Jacobi elliptic function solutions of BBM equation are studied in this paper. Firstly, the knowledge about BBM equation is introduced, and the development and status quo of the equation are fully understudied. Secondly, this paper introduces F-expansion method and relations between values of (P0,P2,P1)and corresponding F((?)) which satisfied F’2=P0+P2F2+PAF4 Thirdly, it is studied that extensional Jacobi elliptic function expansion method and its main steps of solving nonlinear partial differential equation. This expansion method combines the Jacobi elliptic function expansion method mentioned earlier with F-expansion method. Finally, using the Jacobi elliptic function expansion method and the extensional Jacobi elliptic function expansion method to solve the exact solutions of BBM equation with λut+μux+γuux+wuxxx=0. In the process, the application of Matlab greatly simplifies the calculation.3. This paper studies that using the Jacobi elliptic function expansion method and the extensional Jacobi elliptic function expansion method to solve the exact solutions of KP-BBM equation with (γut+μux-γ(u2)x-wuxxt)x+kuyy= 0.4. It also introduces how to solve the exact solutions of Buegers-BBM equation with ut+σ-(u+1)2ux-μuxxt=0 by the Jacobi elliptic function expansion method and the extensional Jacobi elliptic function expansion method.