Algorithmic determination of infinite-dimensional symmetry groups for integrable systems in 1 + 1 dimensions

Abstract

We present algorithms and describe CA-packages to compute the infinitesimal generators of infinite-dimensional symmetry groups for integrable PDEs (evolution equations) in one space and one time dimension. Here, integrable is meant in the sense that the vector field defining the equation is a member of the abelian part of some infinite-dimensional Virasoro algebra. The method of computation is completely different from the usual prolongation method, no determining equations are solved. Instead, all necessary generators of the finitely generated Virasoro algebra are computed from one given element by direct Lie algebra methods. The implementation of the algorithms in MuPAD is described. A sample session is included in which the recursion structures of the KdV and the Krichever-Novikov equations are computed.