Study On The Symmetry Group And Conservation Laws Of Several Nonlinear Partial Differential Equations

Recently,nonlinear models are applied in many fields such as hydrodynamics,option pricings and nerve conductions etc.These models can be described by one or several nonlinear differential equations.They have caught many scholars”attentions to study these differential equations.This paper mainly aims at the applications of symmetry groups and conservation law theory,the power series method and the consistent Riccati expansion method to obtain exact solutions of two types of nonlinear partial differential equations.Chapter 1 introduces backgrounds and the current research of the Lie group theory,conservation laws of differential equations.Chapter 2,we study symmetry groups and optimal systems of one dimensional subalgebra of the generalized foam drainage equation.Then the equation is reduced to the ordinary differential equations by the one dimensional subalgebra.Exact solutions are obtained by those ordinary differential equations.Moreover,the conservation laws of the generalized foam drainage equation are constructed by the multiplier method.In the end,the consistent Riccati expansion method is used to obtain interaction solutions of the foam drainage equation.Chapter 3 mainly studies the power series solution of the convective Cahn-Hilliard equation and analyzes the evolutional procedure of surface under different driving forces.Moreover,the consistent Riccati expansion method is applied to search the exact solution of the convective Cahn-Hilliard equation with the linear convective term.And the graphs illustrate behaviors of the exact solution.