Several Constructive Methods For Multi-wave Solutions Of Nonlinear Evolution Equations

In soliton theory, exponential function method, homogeneous balance method and Hirota’s bilinear method are some important methods developed in recent years for solving nonlinear partial differential equations. These three methods are all constructive methods. For one constructive method, a assumed form of ansatz solution containing undetermined parameters should be given at first for solving nonlinear equations. Then the undetermined parameters in the ansatz solution are determined by substituting the ansatz solution into the given equation and thereby some special solutions of nonlinear equations are obtained. Constructive methods for exact solutions of nonlinear equations don’t rely on the Lax pair and possess simple, intuitive distinctive features so that the solution processes of these constructive methods are more directly. Multi-wave solutions are a kind of interaction solutions of nonlinear partial differential equations, including singular multi-soliton solutions and nonsingular multi-soliton solutions. Investigations on multi-wave solutions is one of the important research topics in soliton theory. In this dissertation, on the one hand the exponential function method and homogeneous balance method are improved and extended so as to construct the multi-wave solutions of complex-coefficient nonlinear partial differential equations and nonlinear variable-coefficient partial differential equations. On the other hand, this dissertation gives a new application of Hirota’s bilinear method for constructing multi-soliton solutions of equations with variable coefficients. The main work of this dissertation includes:Firstly, the introduction section introduces the generation and development of soliton theory simply, as well as several methods for constructing exact solutions of soliton equations. Besides, the main work of this dissertation is briefly outlined.Secondly, an exponential function method for constructing the single-wave solution, double-wave solutions and multi-wave solutions of complex-coefficient nonlinear partial differential equations is proposed by improving the assumed form of ansatz solution. The solution procedure of the proposed method is presented in detail by taking Schr?dinger equation as a exampled.Furthermore, based on the given rational fraction of the variable in the assumed ansatz solution this dissertation improves some steps of homogeneous balance method and thereby put forward the improved homogeneous balance method for constructing multi-wave solutions of nonlinear partial differential equations. In the applications of the algorithm, single-soliton solution, double-soliton solution, three-soliton solution of the Gardner equation with variable coefficients are obtained, from which the general formula N- soliton solution of the equation are summarized.Finally, taking use of a new application of Hirota’s bilinear method in constructing multi-soliton solutions of variable-coefficient Whitham-Broer-Kaup equations, this dissertation obtains single-soliton solutions, double-soliton solutions, three-soliton solutions and the formula of N-soliton solutions. In the solution procedure a valid transformation adopted plays an important role. This adopted transformation has been used to get the bilinear form of the transformed equation conveniently and thereby provides the necessary conditions for using Hirota’s bilinear method.