A New Type Of Soliton Equation With Self-consistent Sources And Its Grammian Solutions

Along with the development of the soliton theory, soliton equations with self-consistent sources (SESCSs) have attracted a special attention, especially constructing and solving SESCSs have become an important and difficult research field in the nonlinear partial differential point of view. This paper mainly studied two kinds of method:source gener-ation method and Pfaffianization procedure, using the two methods to structure different types of the generalized vcKP equation with self-consistent sources, get the Grammian-type solutions of the new coupling system, and finally get the relations between the two methods.The first chapter mainly introduces the background and some preliminary knowledge. First of all, we introduce the background of the soliton equation and SESCSs. Then we focus on introducing the definition and property of Pfaffian.In the second chapter, we will mainly infer the original equations with self-consistent sources on the basis of the Grammian-type solution of the KP equation with variable coefficients. The first to lead a new function by potential transform, the equation is con-verted to bilinear equation to solve out the solution in form of the Grammian. By allowing y-dependence and t-dependence of the integral constants in the diagonal elements of the determinant solutions for the vcKP equation, using source generation procedure to con-struct the generalized variable-coefficient Kadomtsev-Petviashvili equation self-consistent sources and getting its Pfaff solution in the form of Grammian.The third chapter focuses on the Pfaff compatible with the source of the generalized variable coefficient KP equation and its solution. Respectively using Pfaff procedure and source generation procedure to construct different form of coupled system of the general-ized KP equation with variable coefficients, and obtain the relationship between the two methods.The fourth chapter is summary and outlook. Summarized the main content in this paper and looking for the direction and possibility for the future development.