Application Of Hirota Bilinear Method In Solving Soliton Equations

The major contents in this paper include:the Hirota bilinear method, the Wronskian solution of the soliton equations and the Pfaffianization of the soliton solutions. In the first chapter, the history of soliton theory and the methods of soloving soliton equations are presented. In the second chapter, the Hirota bilinear method is introduced firstly, then the major work is focused on the simplified Hiorta method and the KdV equation, the KP equation and the potential KdV equation are solved by this method. The new soliton solutions of the KdV equation and the KP equation are given and the figures of the new 1-soliton solutions are presented by Maple program. In the third chapter, the definition and the related characters of the Wronskian are introduced firstly, then the Wronskian solutions of the potential KdV equation is proved. In the forth chapter, we first introduce the definition and the properties of the Pfaffian, then the general determinants and Wronskians are expressed by the Pfaffians. At last, the potential KdV equation is solved by the Hirota bilinear method, and by means of Pfaffian,we derive the N-soliton solution of the potential KdV equation.