Study On Equivalent Transformation And Exact Solution Of Nonlinear Equation

In this paper,the generalized(Clarkson and Kruskal)CK direct reduction method was applied to the generalized column KdV equation.And the fifth-order variable-coefficient Kawahara equation.Thus the corresponding transformation equation was obtained.Then the two transformed equations are analyzed and solved by using Lie group theory.Baside the exact solution of(3+1)-dimensional KP equation is obtained by using the G'/G-expansion method.Firstly,we found out the connection between the generalized column KdV equation with the corresponding constant-coefficient equations,and the connection with the fifth-order variable-coefficient Kawahara equation with the corresponding constant-coefficient equations.Then we found out all the generating element of constant-coefficient equation,and obtained the group invariant solutions and reduced equations.Then we solved the reduced equations.In the(3+1)-dimension KP equation,the existence of the solution is obtained by G'/G-expansion method.And all G'/G solutions are obtained.In Chapter 1,the generalized column KdV equation is transformed into a constantcoefficient equation by using the generalized CK direct reduction method.And determined the relationship between the solutions of the two equations.Then,the transformed equation is studied by using Lie group theory.And the generating element of the equation is obtained.Then the reduced equation is solved by the auxiliary equation.And the exact solution of the original equation is obtained.Then,we give the conservation laws and adjoint equation of the generalized column KdV equation.In Chapter 2,the generalized CK direct reduction method was employed to study the relation between fifth-order variable-coefficient Kawahara equation and the corresponding constant-coefficient equation.Then we used Lie group theory to analyze the fifth-order constant-coefficient Kawahara equation.Thus,Lie point symmetry and reduction equation of fifth-order constant-coefficient equation are obtained.Then to solve the reduced equation.Furthermore,the adjoint equation and conservation law are given.In Chapter 3,the traveling wave transformation,the homogeneous balance principleand(G'/G)-expansion method are applied to deal with(3+1)-dimensional KP equation.The possibility of existence of(G'/G)-expansion solutions for the(3+1)-dimensional KP equation are discussed in detail.All of the(G'/G)solutions to the equations are obtained.To sum up,the generalized CK direct reduction method can find some equivalent relations between the variable coefficient equation and the corresponding constant coefficient equation.Lie group analysis theory can reduce the high-dimensional nonlinear partial differential equation to one-dimensional ordinary differential equation which is easy be solved.