Study On Some Problems Of Solving Exact Solutions To Nonlinear Evolution Equations

Nonlinear evolution equation is one of the important parts of nonlinear par-tial differential equation. This kind of equation is used to describe the evolution process with time. It usually arises from many fields of science such as physics, chemistry, information science and life science. If the exact solution of a specific equation can be obtained, it will be helpful to clarify the movement of researched object under the nonlinear interaction, it can also interpret various nonlinear phenomena accurately as well as discover new laws of natural phenomena. In recent years, with the development of symbolic computation, solving the exact solutions to nonlinear evolution equations becomes an active research field and many direct algebraic methods for constructing exact solutions are proposed.This dissertation is devoted to analyze some methods in recent years which solving exact solutions of nonlinear evolution equations and study some exact solutions of specific equations.This dissertation consists of the following six chapters.In Chapter1, we briefly introduce the background and history about the related work and view some classic methods seeking exact solutions for the non-linear evolution equations, such as inverse scattering transformation method, Painleve analysis, Backlund transformation method, Darboux transformation method, Hirota bilinear method,and so on.In Chapter2, by improving some key steps in the homogeneous balance method and the extended homogeneous balance method, the modified homoge-neous balance methods (Ⅰ) and (Ⅱ) are proposed. Then, generalized Boussinesq equation, KP equation and MKdV equation are chosen as examples to illustrate that bilinear equations of nonlinear evolution equations can be derived by using the modified homogeneous balance method (I). Furthermore,(3+1)-dimensional Jimbo-Miwa (JM) equation and (2+1)-dimensional KP equation are chosen as examples to show that new auto-Backhmd transformations (BTs) of nonlinear evolution equations can be derived by using the modified homogeneous balance method (Ⅱ). Based on the new auto-BTs, two-soliton solutions of the JM equa-tion and KP equation are given by using e-expansion method.In Chapter3, we investigate some traveling wave solutions with more pa-rameters which obtained by (G’/G)-expansion method, new auxiliary equation method and generally Riccati equation method. Firstly, we prove that the (G’/G)-expansion method is equivalent to the extended tanh function method and there are no new solutions of nonlinear evolution equations can be obtained by the (G’/G)-expansion method. Secondly, we reveal that the14solutions of the new auxiliary equation given by Sirendaoreji and the solutions in original auxiliary equation are the same in waveform and wave velocity, only different in phase. Thirdly, we analyze the27solutions of the generalized Riccati equation given by symbol algorithm of Xie et al in literature, and prove that they are equivalent to the known solutions of Riccati equation.Chapter4deals with two direct algebraic methods of seeking exact solu-tions of nonlinear evolution equations:Sirendaoreji’s auxiliary equation method and the tanh-coth method. We first review some direct algebraic methods of constructing exact solutions for nonlinear equation as well as the general steps. Then the solutions of the auxiliary equation are classified according to its three parameters. This classify gives the relationship among solitary solutions, sin-gular solutions and three parameters. Based on the classify, we can not only modify the exist condition of the third kind of solitary solution for MKdV equa-tion in the literature, but also get abundant exact solutions of the dispersive long wave equations in (2+1) dimensions. Finally, when the balance number m less than or equal to two, we prove that the tanh-coth method is equivalent to the hyperbolic-function method.In Chapter5, we give a new trial function to construct the exact solutions of three important equations and analyze the relationship among these solutions. Among three equations, one is the generalized combination of Burgers equation, Kdv equation, Kdv-Burgers equation and Benney equation, the other two are the generalized Fisher equation and the generalized FitzHugh-Nagumo equation. These solutions, which obtained by using our new trial function, show an inter- esting phenomena: kink type solitary wave solutions and complex value solutions always present together. Based on the phenomena, we prove an important result of the solitary wave solutions of the general nonlinear evolution equations, that is the solution in form of tanhθ and tanh2θ±isech2θ appears in pairs.In Chapter6, a generally projective Riccati equation method is presented to construct doubly periodic solutions (Jacobi elliptic function solution) for the coupled nonlinear equations. First, we introduce a generally projective Riccati equation whose solution involves some Jacobi elliptic functions, then we illustrate that this method can be used to derive unify some solutions by using the Jacobi elliptic sine function expansion method, cosine function expansion method and other Jacobi elliptic functions expansion methods. Second, eight kinds of doubly periodic solutions of the coupled nonlinear Klein-Gordon equations are obtained in a unified way by using the generally projective Riccati equation method.