Perturbation Of Partial Differential Equations Approximate Conservation Laws

Nonlinear phenomena exist in nature and human social fields, such as physics, chemistry, society and economy. With the development of science, the nonlinear systems, describing the above nonlinear phenomena, are increasingly focused on by people, and become one of the important research topics. Con-servation laws play an important role in the study of nonlinear systems. In particular, they are not only useful for linearization and integrability of non-linear partial differential equations and so on, but also contribute to analysis existence, uniqueness and stability of the differential equations'solutions. Many partial differential equations in application depend on a small parameter, which are usually called perturbed (approximate) partial differential equations. Peo-ple analysis the perturbed equations on the basis of the unperturbed equations' theories frame and achieve many efficient conclusions. The definition of approx-imate conservation law was introduced by Baikov and Ibragimov according to the notion of conservation law.This paper mainly studies the approximate conservation laws of perturbed PDEs, and obtains the following results:1. Giving a method to construct approximate conservation laws. This ap-proach defines the standard adjoint system, the approximate adjoint system (I) and the approximate adjoint system (II) of the original system by extending the notion of adjoint equations to perturbed PDEs, then uses the approximate Noether symmetry of the above systems to calculate the approximate conserva-tion laws, which are nonlocal and include auxiliary variable.2. Giving the relationship between the approximate Noether symmetry op-erators of standard and approximate adjoint system (I) as well as the relationship between standard and approximate adjoint system(II).3.Using the approximate Noether symmetries to construct the approximate conservation laws of Euler-Lagrange-type equation utt-uuχχ-uχ2+εut=0.The method in 1 is applied to perturbed wave equation utt-uχχ+ε(umut-au+bup)= 0 and perturbed KdV equation ut-uuχ+uχχχ+ε(u2uχ+cu)=0 to find the approximate conservation laws,which are nonlocal,and become local when replacing auxiliary variable with the function satisfying certain conditions.