Exact Solutions And Rogue Waves For Two Kinds Of Nonlinear Wave Eauation

In this paper, we first introduce the theoretical knowledge of exact solution, and then by using Homoclinic(heteroclinic)breather limit method and Darboux transformation, we study nonlinear partial differential equations and obtain the rational solutions, soliton solutions, breather solutions, rogue waves. we verify the relationship between rogue wave and effective wave by practical application. At last, we study stochastic partial differential equation by Fan-algebra method and Truncation expansion method and gain several kinds of white noise functional so-lutions, at the same time, we experience the transformation from stochastic equa-tion to deterministic equation and analysis the relationships of different methods, different solutions.This paper is organized as follows:In the first chapter, the soliton theory, the development of rogue wave and theoretical knowledge of stochastic partial differential equations are introduced.In the second chapter, Homoclinic(heteroclinic)breather limit method is in-troduced firstly, and then using the homoclinic(heteroclinic)breather limit method, we obtain the rational solutions of (3+1)-dimensional KdV equations, and rogue wave.In the third chapter, we first introduce Darboux transformation, and then solve the variable coefficient nonlinear Schrodinger equation by this method and obtain single soliton solution, two-soliton solution, breather. Finally, we get rogue wave by Taylor expansion of breather solution.In the fourth chapter, by applying Fan-algebraic method and Truncation expansion method respectively, we get white noise functional solutions of the Wick-type mixing KdV equation. By contrasting and analysing these results, we find that the effective method for solving stochastic differential equation is Fan-algebra method.