Exact Solutions And Conservation Laws Of Two Kinds Of Nonlinear Evolution Equations

Nonlinear evolution equation is a subject with a long history,which has been widely applied in many aspects such as physics,chemistry,fluid mechanics,ecology and medicine Mathematics'research on nonlinear evolution equations is mainly reflected in two aspects one is about the solution of nonlinear partial equations,such as rational solution,soliton solution and so on.The other is to study the algebraic and geometric properties of integrable systems,such as Hamiltonian structures,conservation laws.In this paper,three problems of the nonlinear evolution equations are studied by using computer algebra as auxiliary tool They are exact solutions,residual symmetry and conservation lawsThe main contents of this thesis are as followsFirstly,we briefly introduce the research background and current situation of interaction solutions,residual symmetry,conservation laws of nonlinear evolution equations.And we specifically introduce some definitions and theorems used in this paperSecondly,we use the compatible Riccati expansion(CRE)method to study the(1+1)-dimensional classical Boussinesq-Burgers system,and we also use the CRE solvability to obtain the soliton solution and some interaction solutions of this system.Combining the solutions of the Riccati equation with the Jacobi elliptic function,we figure out three interaction solutions of the(1+1)-dimensional classical Boussinesq-Burgers system.We also draw the corresponding waveform diagram by using the mathematical software mapleLastly,we use the truncated Painleve method to obtain the residual symmetry of the Kaup-Boussinesq equation.At the same time,the Kaup-Boussinesq equation are proved to be compatible Riccati expansion(CRE)solvable.With the help of the CRE method and Riccati equation,we can obtain the interaction solution of the Kaup-Boussinesq equation.Then by using Lie group analysis method to study the Kaup-Boussinesq equation,the Lie point symmetries of this equation are obtained.And it is demonstrated that the Kaup-Boussinesq equation are nonliear self-adjointness.This property is applied to construct infinitely many conservation laws of the Kaup-Boussinesq equation with the corresponding Lie symmetry by Ibragimov thereom.