Invariant Subspace And Lie Group Analysis For Several Kinds Of Nonlinear Partial Differential Equations

In this paper,the RLW-KdV equation,the fifth-order dispersion equation with variable coefficient and the Kudryashov-Sinelshchikov equation are analyzed.Firstly,all vector fields,single-parameter transformation groups,group-invariant solutions,adjoint equations and the conservation law of the first equation are obtained by Lie group analysis.Secondly,the Lie group analysis is used to classify the variable coefficients of the second equation,and the specific reduction equation is obtained.The explicit exact solutions and conservation laws of the equation are obtained.Finally,several different forms of exact solutions for the third equation is constructed using the invariant subspace method.In the first part,the classical Lie group theory is used to analyze and study the non-linear RLW-KdV equation.The vector field of the equation is obtained,the invariants of the equation are obtained,and the ordinary differential equation is reduced by the group invariant solution.The trigonometric function solution,hyperbolic function solution,rational function solution and exponential function solution of the original equation can be obtained by the e^-??x? expansion method and Lambert W function method.Finally,we divide the equation by Lie group.The adjoint equation,Lagrangian quantity and conservation law of the nonlinear RLW-KdV equation are obtained by analysing the vector field.In the second part,the classical Lie group theory is used to study and analyze the fifth-order dispersion equation with variable coefficients.The relationship between the undetermined coefficient function and the variable coefficients must be satisfied is obtained.The selection of variable coefficients is classified according to symmetric constraints.All vector fields and the specific fifth-order dispersion equation with variable coefficients are obtained.The reduced equation can be obtained by using invariants.The exact solution of the equation is obtained by the power series expansion method,the exponential function method and expansion,the adjoint equation and conservation law of the fifth order dispersion equation with variable coefficients are given.In the third part,the invariant subspace method is used to obtain the exact solution of Kudryashov-Sinelshchikov equation.The main idea is that the invariant subspace allowed by the nonlinear partial differential equation?group?is constructed by solving the subspace obtained by the linear ordinary differential equation.The invariant subspace allowed by the nonlinear differential operator in the Kudryashov-Sinelshchikov equation is studied and given.The method of invariant subspace is used.Exact solutions in the form of polynomials,trigonometric functions and exponential functions can be constructed.