Characteristic Set Algorithm for Determining Approximate Symmetries of Perturbed Partial Differential Equations

Abstract

In this paper, based on differential characteristic set algorithm of a differential polynomial system, an algorithm for calculating two kinds of approximate symmetries of a per- turbed partial differential equation is suggested. The difficulty of solving determining equations of the approximate symmetries is overcome significantly by the algorithm. As applications of the algorithm, the approximate (potential) symmetries of three evolution equations are determined. This is also a new application of differential characteristic set (Wu's) method in differential field. I. INTRODUCTION The two kinds of approximate symmetry methods as ex- tension of classical symmetry method for analyzing partial differential equations (abbreviated to PDEs) with small pa- rameters have been developing(1)-(6) for decades. One of the methods was firstly developed by Baikov, Gazizov and Ibragimov, which represents a perturbation technique embed- ded into the standard procedure of classical symmetry method (1), (2). Another approach was developed by Fushchich and Shtelen and later followed by Euler et al. which combines perturbation technique with the classical symmetry method (5), (6). However, some algorithmic difficulties are encountered in calculating these approximate symmetries of PDEs. One of the main difficulties is solving determining equations (abbreviated to DTEs) of the symmetries. So far, no efficient method in dealing with this difficulty has been found (7). The purpose of this article is to suggest an algorithm based on differential characteristic set (abbreviated to dchar-set) algorithm (8), (9) to determine the approximate symmetries of nonlinear perturbed PDEs. The difficulty mentioned above is overcome efficiently by this algorithm. In this section, we first give the outline of main procedures of the two kinds of approximate symmetry methods, then introduce the main problem to be solved in this article.