The Solution Method Of The Variable-Coefficient Korteweg-De Vries Equation (vcKdV) And The Variable-Coefficient Kadomtsev-Petviashvili (vcKP) Equation

With the development of the Nonlinear Science, lots of Nonlinear Evolution Equations (NLEEs) are found, and they are playing important roles in many different physics fields. Solitons as an important branch of the Nonlinear Science, have been well studied and widely applied, which is of important physics significance. Up to now, there are many kinds of methods to obtain the solutions of the NLEEs such as traveling wave method, Ba|¨cklund transformation method, Hirota method, homogeneous balance method, Wronskian method and Pfaffian method. This paper is precisely take the NLEEs as a foundation , and study several kinds of significant methods to solve the NLEEs, at the same time, can extract the new solutions of a variable-coefficient Korteweg-de Vries (vcKdV) equation and a variable-coefficient Kadomtsev-Petviashvili (vcKP) equation.The following we introduce the basic contents of this paper:In chapter one, we first introduce the history and development of the soliton, and then by means of several examples explain three methods—traveling wave method, Ba|¨cklund transformation method and homogeneous balance method. In chapter we study the Hirota method, which is developped in 1970s. We introduce the D- operator, the special properties of the D- operator and the transformation. As an example, we find an exact solution for the KdV equation.Chapter three focus on the Wronskian technique. This technique profits from the special structure of a Wronskian determinant, which can contribute simple forms of its derivatives. Through the properties of the Wronskian, we obtain the Wronskian solution of the KP equation. Then, with the help of the Wronskian technique and the method of the solving the KP equation, we derive the Wronskian determinant solution for a variable-coefficient KdV equation and variable-coefficient KP equation.Chapter four as the focus of this paper in which we study another determinant solution for nonlinear evolution equations is the Grammian determinant. We proved that the bilinear KP equation could be reduced to a Pfaffian identity by taking its solution as a Grammian determinant. When the Grammian solution is expressed as a Pfaffian, the bilinear equation is equivalent to the Pfaffian identity by virtue of Pfaffian derivative formulation. In this chapter, we first give the definition of the Pfaffian and the Pfaffian identity. Then, we use the method to solve the variable-coefficient KP equation and obtain the Grammian solution.