Constructing Exact Solutions Of Nonlinear Partial Differential Equations Via Painleve Method

Exactly solving nonlinear partial differential equations has both theoretical significance and application value. Because of the complexity of the nonlinear partial differential equation itself has, there are still a lot of important equations cannot been found out exact solutions. Although some equations can be solved, the types of the obtained solutions are relatively single. This shows that seeking new exact solutions of nonlinear partial differential equations is a meaningful research work. Painleve method is such a method which sets together of distingushing the integrability and exactly solving solution of nonlinear partial differential equations. Since 1983 when Weiss, Tabor and Carnevale proposed the Painleve test of partial differential equations, using the method to discuss the integrability of partial differential equation or exactly solving solutions has made great progress but it is still worthy of further investigation. This paper takes high-dimensional nonlinear partial differential equations, nonlinear partial differential equation with variable coefficients and the system of nonlinear partial differential equations as the research target to apply Painleve analysis so as to conclude that what constraint conditions should satisfy when equation(set) has Painleve property. At the same time, new exact solutions of the equation(s) are obtained by using the Painleve truncated expansion. And some figures of partial solutions are plotted for analyzing the property of the solutions. Firstly, this paper briefly summarizes the background and history of development of the soliton theory, summarizes and deduces the Painleve analysis method. Then on the one hand the WTC method of Painleve analysis is used to test the integrable properties of the(4+1)-dimensional Fokas equation, variable-coefficients KdV equation in fluid and the(1+1)-dimensional Broer-Kaup(BK) equations.On the other hand, new solutions of Fokas equation, variable-coefficients KdV equation in fluid and the(1+1)-dimensional Broer-Kaup(BK) equations are obtained by using Painleve truncation expansion. At the same time, the evolutionary characteristics of some typical solutions are analyzed.