Application Of Bilinear Method To Some Wave Equations

In this dissertation, we mainly study Hirota bilinear method to some aspects in soliton equations, especially the aspect of finding exact solutions. The topics include several aspects: Constructing and solving the variable-coefficient KP equation and its integrability, such as Backlund transformation, nonlinear superposition formula and so on; Derivation of the N-soliton solutions in terms of Wronskian and Grammian determinant for the variable-coefficient KP equation; Generalization of Hirota method to study new exact solutions of some wave equations, namely periodic wave solutions.To find exact solutions to soliton equations is an important aspect in soliton theory. We commence the investigation on two aspects: solutions expressed in term of Hirota, Wronskian and Pfaffian and construction of exact solutions. It is shown that the bilinear method provides a very powerful tool in searching for exact solutions of soliton equations. It also relates the uniformity of soliton equations with the diversity of methods, such as Hirota bilinear method, Wronskian technique and Backlund transformation. Accordingly, it excavates more essential attribute of soliton. The specific work consists of three parts:(1). In chapter 2, we first present the definition and elementary properties of the bilinear operators. Next, we derive the N-soliton solutions for the variable-coefficient KP equation in detail. Then, the Backlund transformation, nonlinear superposition formula and Lax pair of the variable-coefficient KP equation are constructed. (2). In chapter 3, we first discuss the solutions of soliton equations expressed in term of Wronskian, including the concerned properties of Wronskian and. construction of the Wronskian determinant solution to the variable-coefficient KP equation. Second, we give the elementary properties of Pfaffian and obtain the Grammian determinant solution of the variable-coefficient KP equation. It is noted that the solutions in the process are expressed by Grammian-type Pfaffian.(3). In chapter 4, it consists of three sections. In the first section, we generalize the Hirota method and present new exact solutions of the Boussinesq equation, namely periodic wave solutions. Moreover, the properties of the solutions are described by figures. In the second section, we derive periodic wave solutions of the two-dimensional Boussinesq equation by means of the aforementioned method and obtain its rational solutions by taking the long wave limit on soliton solutions. A summary and discussions are given in the last section.