Soliton And Integrable Systems Based On Bilinear Method

The search on the explicit solution and integrability of nonlinear evolution equations are helpful in clarifying the underlying algebraic structure and play an important role in reasonable explaining of the corresponding natural phenomenon and application.In this dissertation, based on the Hirota bilinear method, the exact solution and physical problem are systematically investigated and the more abundant structure in bilinear formalism are revealed as well as the relations to other direct methods and integrability are discussed. It is evident that Hirota bilinear method is a powerful tool for solving a wide class of nonlinear evolution equation. More remarkable is that various physically important solutions to the soliton equations can be presented explicitly by means of Hirota bilinear method. This dissertation may be divided into two parts.Part I is devoted to summarize some properties of Hirota's bilinear operators, the properties of Wronskian and Pfaffian that appears in the expression of the N-soliton solution of the soliton equation. It is shown that the solutions of most of the soliton equations are given in terms of the determinants with a Wronskian and Grammian structure, or Pfaffian. The bilinear forms of the equations are reduced to the algebraic identities for determinants or the identities of Pfaffian.By using Hirota bilinear method and Wronskian technique, we consider the iV-soliton solutions of the nonisospectral MKdV equation and the MKdV equation with nonconformity, respectively.In addition, soliton equations whose solutions are expressed by Pfaffian are briefly discussed. Included are KP equation, BKP equation and theirs Backlund transformation in bilinear form. By applying the pfaffianization technique to the soliton equation, a new integrable model with Pfaffian solution could be generated.Part II is mainly focused on studying and the construction of various type of explicit exact solutions for nonlinear evolutions on the basis of bilinear formalism.The Hirota bilinear method is generalized and investigated, where the N-soliton-like solutions with singular slowly decaying at infinity for the shallow water waves equations and Ito equation are obtained. Further, we also explored the interaction behaviors and find singularity.