Research On Exact Solutions Of Some Nonlinear Evolution Equations

Nowadays, as one of the most important research subject of nonlinear systemswhich almost touches every field, soliton theory is of great value in scientific researchesand applications to explain the wave propagation and natural phenomena as well asto determine the physical attributes of materials. One of the main contents of solitontheory is to seek the solution of nonlinear system, especially the solitary wave solutions(including exact solutions and numerical solutions).Over the past decades, the study on soliton theory of nonlinear evolution equation,especially the study on solitary wave solutions, developed quickly and many methodson the solitary wave solutions of the nonlinear evolution equation were proposed. Re-cently, with the development of mathematics mechanization, the study on solitary wavesolution becomes more and more dependents on computers., which results in a seriesof new methods are obtained and used to study the discrete solitary wave solution ofnonlinear differential-difference systems and the stochastic solitary wave solution ofnonlinear stochastic systems. These methods have been the main content of studyingon nonlinear evolution equations.In chapter1, we introduce the background and development of mathematics mech-anization and soliton theory, and summarize and analyze the methods known for solvingthe nonlinear evolution equation. The significance and contents of this research are alsoincluded in Chapter1.In chapter2, F-expansion method and extended F-expansion method are intro-duced. Then by using the methods, the exact solutions expressed by Jacobi ellipticfunctions for the Davey-Stewartson equation are derived. In the limit cases, the solitarywave solutions and some types of traveling wave solutions for the system are obtained. In chapter3, Tanh function method to solve nonlinear differential-difference e-quation is discussed. Then we apply the method to the study of the (2+1)-dimensionalToda lattice, lattice equation, discrete Scho¨dinger equation with saturated nonlinearityand Ablowitz-Ladik lattice model. By comparing with (G′G)-expansion method whichis popular recently, we draw a conclusion.In chapter4, the related theory of stochastic differential equation is introduced andTanh function method to solve stochastic differential equation is presented. Throughthe method, we obtain three different types of solutions to generalized stochastic KdV-Burgers equation. Finally we apply F-expansion method to study generalized stochasticKdV equation set.In chapter5，the summary and prospect are given.