Application Of Wronskian Technique To Some Soliton Equations

This dissertation mainly consists of two parts. The first part is to present Wronskian and Grammian solutions to some soliton equations and give the proofs. In the second part, the balance method and the refined invariant subspace method are applied to generate exact solutions to nonlinear evolution equations.The first chapter summarizes the formation and development of the soliton theory.The second chapter introduces some concepts, formulas, properties used in this the-sis, such as Hirota operator, Wronskian and the refined invariant subspace method.In chapter three, we derive the Wronskian and Grammian solutions of the gener-alized variable-coefficient (3+1)-dimensional and (n+1)-dimensional KP equation. In addition, three different kinds of Grammian conditions consisting of linear partial differ-ential equations system are derived, which guarantee that the Grammian solves a (3+1)-dimensional generalized shallow water equation in the Hirota bilinear form.The fourth chapter consists of two sections. In the first section, the balance method is firstly presented to derive the N-th order Wronskian solutions and Grammian solutions of nonlinear evolution equations, by which the Wronskian and Grammian solutions of the (2+1)-dimensional KP equation and the (2+1)-dimensional KdV equation are obtained. In the second section, the refined invariant subspace method is applied to construct exact solutions to nonlinear evolution equations.