Nonlinear Perturbation Equations Approximate Noether Symmetry And Approximate Conservation Law

In the real world, many physical phenomena can be described by partial differential equations (PDEs). There is such a class of PDEs with relatively small parameters, we call those equations the perturbed equations, which differ from the unperturbed ones. Therefore, the study of perturbed equations is of great significance. In the fields of systems of differential equations as well as their applications, the conservation law plays a crucial role. Consequently, the construction of approximate conservation laws for perturbed equations is an important issue. The development of methods for constructing conservation laws for the unperturbed PDEs will greatly promote the development of methods for constructing approximate conservation laws for the perturbed ones. A. H. Kara extended the partial Noether method of constructing conservation laws to that of construction of approximate conservation laws for perturbed equations. One can construct the approximate Noether symmetries and approximate conservation laws for linear perturbation equations by the partial Lagrangian.However, the more interesting question about how to construct approximate conservation laws for the more general non-linear perturbation equations is naturally raised. Thus, on the basis of previous works, in this paper we correct the incorrect definition in literature and redefine the concept of partial Lagrangian theoretically. Moreover, we construct the approximate Noether symmetries and approximate conservation laws for variety of types of non-linear perturbed equations, and express the results in general forms. More importantly, we obtain complete classifications of approximate conservation laws for different forms of perturbed equations. This greatly extends the use of the method, and verifies the usefulness of the method as a tool.The content of this paper is divided into four chapters. The first part is a brief review of the history of researches on the conservation laws and approximate conservation laws and the background and significance of the subject relating to the topics of this article, we also outlines the main results of this paper. In the second part we present the theory necessarily relating to approximate conservation laws. In the third and fourth parts, respectively, we construct the approximate conservation laws and conserved vectors for the nonlinear perturbed wave equation and the perturbed sine-Gordon equation by the latest partial Lagrangian function method, and obtain complete classifications of their approximate conservation laws.