Painlevé Analysis,Symmetries And Exact Solutions Of Nonlinear Partial Differential Equations

In the process of solving exact solutions of nonlinear partial differential equations.Many scholars have found that the exact solution is of great significance in both theoretical research and practical application.Through further research,it is found that the Painlevéanalysis method can not only distinguish the integrability of nonlinear partial differential equations,but also find the exact solution of the given equation.In this paper,the KP equation and Hirota-Satsuma equations are studied by using Painlevé method.In chapter 1,KP equation is studied.The Painlevé method was used to analyze the equation.First,the integrability of the equation was verified by the WTC method of Painlevé analysis.Secondly,the new exact solution of the equation is obtained by the truncated expansion method.In the process of solving,the evolution characteristics of partial decomposition can be analyzed.In chapter 2,the Hirota-Satsuma equations are analyzed and studied.First,the prime analysis of the equations is carried out.Second,the Painlevé expansion of the equations is determined,and the compatibility conditions of the original equations are determined by finding the resonance points.Finally,the exact solution of the system is obtained.Through the research,it is found that Painlevé method is not only applicable to the exact solution of a single equation,but also applicable to the exact solution of equations.In chapter 3,the(2+1)dimensional CGKP equation is studied by using Lie group analysis,Firstly,the generator of CGKP equation is obtained by using the method of vector field extension.In the process of solving,the single-parameter transformation group corresponding to(2+1)dimensional CGKP equation is obtained.Secondly,the(2+1)dimensional CGKP equation is simplified to the(1+1)dimensional partial differential equation,and then the(1+1)dimensional CGKP partial differential equation is simplified to the ordinary differential equation.Finally,the exponential function is used to solve the reduced equation,and the exact solution is obtained.At the end of this chapter,the adjoint equation and conservation law of(2+1)dimension CGKP equation are obtained.To sum up,the WTC method of Painlevé analysis is firstly applied to KP equation.Secondly,it is generalized to the Hirota-Satsuma equations and the exact solution of the equations is obtained.By applying Lie group analysis to(2+1)dimensional CGKP equation,the(2+1)dimensional nonlinear partial differential equation is reduced to a one-dimensional ordinary differential equation which is easy to solve through symmetry and similarity reduction,and the exact solution of the equation is obtained.In addition,the conservation law of(2+1)dimension CGKP equation is also constructed.