The Jacobi Elliptic Function Solutions For Several Nonlinear Partial Differential Equations

The objectives of this work are twofold. Firstly, a mathematical techniquebased on auxiliary equations and the symbolic computation system Matlab isdeveloped to construct the exact travelling wave solutions to a Klein-Gordonequation and a generalized Benjamin-Bona-Mahony equation. Secondly, severaltypes of traveling wave solutions for two nonlinear partial di?erential equationswith variable coe?cients are derived by using the auxiliary di?erential equationtechnique.Chapter 2 deals with a nonlinear Klein-Gordon equation u_tt -a²u_xx +αu-βuⁿ = 0, some new solutions including the Jacobi elliptic function solutions,the degenerated soliton-like solutions and the triangle function solutions to theequation are obtained.In chapter 3,by making use of an auxiliary equation (dd/zξ)₂ = c₁+c₂z²+c23 z4,we study a generalized Benjamin-Bona-Mahony equation ut + aux - buxxt +k(um)x = 0. The Jacobi elliptic function solutions, the degenerated soliton so-lutions and the triangle function solutions to the equation are obtained undercertain circumstances.In chapter 4, we study two nonlinear partial di?erential equations withvariable coe?cients ut + a(t)u_x + b(t)(uⁿ)_x + k(t)(uⁿ)_xxx = 0 and ut + a(t)u_x +b(t)(uⁿ)_x + k(t)(uⁿ)_xxt = 0. With the help of the symbolic computation systemMatlab, many new exact travelling wave solutions of the two equations, includ-ing solitary wave solutions and triangular solutions, are obtained.