Soliton Theory And Exact Solutions For Nonlinear Evolution Equations

This paper mainly study some nonlinear evolution equations with physical background by using the homogeneous balance method and the tanh method, based on the soliton theory and method. With the modern computer technique, we find their new soliton solutions and other new exact solutions. Furthermore, we generalize the tanh method.At first, by using the Riccati equation v' = bx + b2v2 as the disturbance equation, we construct the new soliton solutions for a series of nonlinear evolution equations, including1 Reaction and Diffusion Equation:where are parameters. 2 Coupled Klein-Gordon-Schrodinger Equation(KGS):3 Nonlinear Wave Equation:where a1 a2 a3 a4 are parameters.4 Generalized Bergurs-BBM Equation (B-BBM):where are parameters.We get not only their old soliton solutions but also a kind of shock wave soliton solution, trigonometic function periodic solutions and rational solutions. By using the Matlab, we obstain their numerical simulation pictures.Then, with the new assumption on the solution ,we change the disturbance equation with v' = k(\-v2) and reconstruct the soliton solutions of the reaction and diffusion equations and the nonlinear wave equations. We find that they all have a kind of complex linear soliton solution. By using the Matlab, we find that the real part of the solution is a kink soliton and the imaginary of it comprise a bell soliton.In the following section, we reconstruct the soliton solution of KGS equation wholly by use of the disturbance equationwhere are parameters.Through varying the parameters in the disturbance equation, we get abundant types of soliton solutions of the KGS equation. Finally, we study the (2+l)-dimension nonlinear dispersive long wave equationSkillfully, we find the self-transformation between the two equations and turn the (2+l)-dimension nonlinear dispersive long wave equation into a simple (2+l)-dimension PDE. Then, with the homogeneous balance method we obtain the Hopf-Cole transformation between the (2+l)-dimension PDE and the heat equation. Through it, we obtain abundant exact solutions of the (2+l)-dimension nonlinear dispersive long wave equation with the Matlab, including the multi-solitary wavesolutions.