Computer Program Software for Determining formal Symmetry of Evolution Eqations

Abstract

The existence of formal symmetry of an evolution equation is one of the criteria of the complete integrability or solvability of evolution equations, due to Sokolov and Shabat. Many evolution equations such as the soliton (solitary equation) of Korteweg-de Vries (KdV) equation have been found recently to have various kinds of explicit integral or solutions. Such evolution equations admit infinitely many symmetries or admit the recursion operator. In this paper we introduce the definition of the formal symmetry. Formal symmetry is the approximation of the recursion operator, which brings us to a convenient way of characterizing equations admitting infinitely many symmetries. In this research, we developed a program for computing the formal symmetries of evolution equations. To verify the correctness of the program, we apply it to some evolution equations (as testing equations), which have been proved to be formally completely integrable. The program we obtained can compute the formal symmetry of finite arbitrary order (up to order 18) of the testing equations, which verify the correctness of the program