Exact Solutions For (3+1)-dimensional Nonlinear Evolution Equation And Variable-Coefficient KP Equation

In the aspect of theory, exact solutions of soliton equations can help us to understand the algebraic structure and basic properties of the equations. In the actual application, exact solutions of soliton equations can explain some related natural phenomenon. Pfaffianization technique is a kind of effective means for solving soliton equations, this technique solves many spectrum equations, the discrete system and soliton equations. So the research that the application of Pfaffian techniques is exact solutions of the soliton equations, is always the important topic what mathematicians and physicists are concerned. This paper summarizes the methods for calculationing the exact solutions of constant coefficient and variable coefficients soliton equation.1. The first and second chapter, it mainly introduces the produce and contents of the soliton theory and the research status and development of soliton equation. It gives to three methods of solving soliton equations.2. The third chapter studies the exact solutions of the (3+1)-dimensional nonlinear evolution equation. Firstly, under the conditions of the Hirota bilinear operator and Cole-Hopf transformation, nonlinear evolution equation turn into linear form. With the related property of Wronskian determinant prove Wronski-type pfaffian solution for the equation. Secondly, using the Pfaffianization works out the coupled system of (3+1)-dimensional nonlinear evolution equation and Gramm-type Pfaffian solution. Finally, in the linear partial differential conditions and extend, under a set of linear partial differential condition and the extending linear partial differential condition, we obtain the Pfaffian solutions and the extended Pfaffian solutions to (3+1)-dimensional nonlinear evolution equation. By constructing a proper relationship, we gives N-soliton solution for the (3+1)-dimensional nonlinear evolution equation.3. The fourth chapter studies the exact solution of the (3+1)-dimensional variable coefficients KP equation. Firstly, because the nonlinear evolution equations is constant coefficient equation, using the logarithmic transformation and rational transformation, u will set into the form of variable coefficient. It can be that (3+1)-dimensional variable coefficient KP equation turns into bilinear form. Then, using the Wronskian technology and Pfaffianization method, we solve the coupled system of a (3+1)-dimensional variable coefficients KP equation. Finally, y, z, t set into the special linear partial differential conditions, we obtain the Pfaffian solutions and the extended Pfaffian solutions to a (3+1)-dimensional variable coefficients KP equation, and accurately calculate the two-soliton and three-soliton solutions. By constructing a proper relationship, we gives N-soliton solution for (3+1)-dimensional variable coefficients KP equation.