Cauchy Problem For Nonlinear Evolution Equation And Symmetry Reduction

The nonlinear phenomena widely appear in almost all the fields such as na-ture, society, economy and so on. With the development of science, people pay more attention to the nonlinear phenomena which reflect natural phenomena, so the researches on the nonlinear systems are making more and more progress. It is well-known that the equation of mathematical physics play a key role in the physics and many scientific fields. A lot of mathematician and physicist try to obtain the solutions of differential equations, especially for construct-ing exact solutions of nonlinear partial differential equations. As is known that symmetry group plays an important role in the constructing solutions of PDEs. Symmetry group method pioneered by S.Lie is the most effective and universal method for constructing exact solutions of these equations. We refer to it as classical method. However, the symmetry groups admitted by some problems arising in applications are not rich enough for one to use the technique of symmetry reduction, especially for the analysis of boundary and initial-value problems. In order to solve these problems, various generaliza-tions of the classical method have been developed. These include conditional symmetry, generalized symmetry , higher conditional symmetry, etc.This paper is organized as follows: In section 1, we provide some back-ground material and notations of GCS and theory of symmetry reduction. In section 2, we discuss the classification and symmetry reduction for the general-ized KdV-type equations u_t = G(u)u_xxx + F(u, u_x) by higher conditional sym-metry method. In section 3, we pay attention to a class of fourth-order PDEs u_t = -u_xxxx - H(u)u_xx² + F(u)u_xx + Q(u)u_x² + R(u). We classify the equations admit high-order generalized conditional symmetries and reduce the resulting equations to Cauchy problems for some systems of first-order ODEs. We can construct analytic solutions of the initial-value problems for PDEs, these solu-tions generally cannot be obtained via the classical symmetry method or the conditional symmetry method, section 4 is the conclusions and the study in the future.In this article, we have developed the theories to single higher-order equa-tions. The solutions which we obtained can give more useful information to the corresponding equations.