Some Methods For Finding Explicit Solutions Of Nonlinear Evolution Equations

With the development of science and technology, many new nonlinear evolution equa-tions(NLEEs) are found in some nonlinear system. In other handy some famous NLEEs are discovered in some new fields. The NLEE based on physics is one of the significant subjects in contemporary study of nonlinear science. Exploring and developing new method to solve the NLEEs is the forefront topics in the studies of nonlinear physics problems.In recent years, many powerful methods for constructing the solution of NLEE are proposed. Particularly, the development of symbolic computation and mathematics mechanization accelerate the research of NLEE greatly. In this paper, 225 kinds of NLEE were collected. Some new methods for searching approximate and exact solutions of NLEEs are extended and modified, meanwhile, we apply it to new fields of physics. As a result, many helpful conclusions are obtained.In studying for the approximate solution of NLEE, our work consists of the following four parts. The self-similar translation based on Lie group method is applied to Garnder equation. By using reductive perturbation theory, we analysis nonlinear NLS equation. The power series expansion method is utilized to study the collective behavior in dusty plasma consisting of cold dust particles and two-temperature isothermal ions. For the cylindrical and spherical KdV equation which has been described the nonplanar dust ion-acoustic solitary waves, we obtain the adiabatic approximation solutions by using the equivalent particle theory.In studying the exact solution of NLEE, our work consists of the following two parts.Firstly, by studying the Liu's Jacobi elliptic function expansion(JEFE) method and extended JEFE method of Yan, we further extended and modified these method. With the aid of symbolic computation, we can not only successfully recover the previously known formal solutions but also construct new and more general formal period solutions(including solitary wave solutions) for some NLEEs, such as mBBM equation, Gardner equation, Ito's 5th-order mKdV equation and (3+l)-dimensional KP equation and so on. Secondly, based on the Lame function and JEFE method, the perturbation method is applied to some NLEEs, such as NLS equation, Zakharov equation, (2+l)-dimensional ZK and mZK equation and so on. Many new multi-order envelope periodic solutions and multi-order periodic solutions are derived to these NLEEs.It is worthwhile to mention that the above methods can also be applied to some other NLEEs.