The Study Of Invariant Solutions And Conservation Laws For Several Classes Of Nonlinear Evolution Equations

Nonlinear evolution equation(Group)has important practical significance to explain complex phenomena and solve problems in the field of physics,chemistry and other fields.It can not only carry on the quantitative research on the problem,but also provide a necessary basis for the practical problems of qualitative theoretical analysis.So the problem of solving the nonlinear evolution equation(Group)is one of the most important research topics in this field.In this paper,by using the classical Lie group method,the extended tanh function expansion method,the method of Riccati auxiliary equation and the power series expansion method,several kinds of nonlinear evolution equations to be solved,such as a class of KdV-mKdV equation,the extended(2+1)-dimensional Jaulent-Miodek equation,the extended Kadomtsov-Petviashvili-Benjamin-Bona-Mahoney(KP-BBM)equation,(3+1)-dimensional Zakharov-Kuznetsov-Burgers(ZKB)equation.A large number of new exact solutions and the conservation laws of the equations are obtained by solving the reduction equations(Group).In chapter one,the homogeneous balance method and the(G?/ G)-expansion method are used to obtain the solutions of the KdV-mKdV equation under different conditions.The results show that the calculation of the(G?/ G)-expansion method and the homogeneous balance method is simple and accurate,and it is also applicable to other nonlinear evolution equationsIn chapter two,by applying the direct symmetry method,and with the help of the computer algebra system Maple,a lot of new solutions of the extended(2+1)-dimensional Jaulent-Miodek equation are derived,which include Weierstrass periodic solutions,elliptic periodic solutions,rational function solution and so on.At last,the conservation laws of the equation are obtained based on the symmetry and adjoint equations.In chapter three,based on the classical Lie group methoud,we obtained the Lie point symmetries and group invariant solutions of the extended Kadomtsov-Petviashvili-Benjamin-Bona-Mahoney equation.A great many of solutions are derived by solving the reduction equations with the auxiliary function method.And the conservation laws of the equation is given at last.In chapter four,by using the undetermined coefficient method,the classical Lie symmetry and reduced equation of the(3+1)dimensional Zakharov-Kuznetsov-Burgers equation are obtained.At the same time,a great many of new explicit solutions are derived by using the power series expansion method and the method of Riccati auxiliary equation.Finally,on the basis of the symmetry and conjugate equation,the conservation law of the equation is obtained.In summary,the main characteristics of this paper is to apply the classical Lie group theory to the process of solving the nonlinear partial differential equations(Group).We get symmetry and reduction equation by using the classical Lie group method and the undetermined coefficient method.In order to achieve the purpose of reducing dimension and easy to solve the equations,we choose a proper transformation,and the reduced equation is discussed by means of different auxiliary equation method.