The Symmetry Of The Nonlinear Pde, Disturbance And About,

In this paper, we discuss mainly two aspects about symmetry method to solve nonlinear PDEs and perturbed nonlinear PDEs, and they read as follows: Firstly Lie-B(?)cklund symmetry method is developed to solve the perturbed nonlinear evaluation equation. A central object of the approach is an integrable unperturbed equation, which is constructed by defining a proper Lie - B(?)cklund group of transformations and applying it to lead to the perturbed equation. This method is a generalization of Lie point symmetry technique. We consider the perturbed KdV equation as an illustration of the approach. New approximate solutions of the perturbed KdV equation stemming from the exact solutions of the unperturbed equation are obtained using the Lie-Backlund symmetry method. These approximate solutions of the perturbed KdV equation are the soliton-like solutions. Secondly, we show that for some class of nonlinear partial differential equations with arbitrary order the determining equations for the nonclassical reduction can be obtained by requiring the compatibility between the original equation and the invariant surface condition. The nonlinear wave equation, the Boussinesq equation and the nonlinear Klein-Gordon equation all serve as examples illustrating this fact. More, we show that for very general (1+1) dimension nonlinear partial differential equation with arbitrary order the determining equations for the nonclassical method can be derived by the compatibility between the original equation and the invariant surface condition. Then we generalize this result to the system of the (m+1) dimension partial differential equations. So the determining equations for nonclassical reduction method of the system of the differential equations, can be constructed by simply imposing compatibility between the original system and the invariant surface conditions, instead of computing the coefficients of the extension of infinitesimal generator, which are very complicated for the (m+1) dimension partial differential equation.