Additional symmetries for integrable equations and conformal algebra representation

Abstract

We present a regular procedure for constructing an infinite set of additional (spacetime variables explicitly dependent) symmetries of integrable nonlinear evolution equations (INEEs). In our method, additional symmetry equations arise together with their L-A pairs, so that they are integrable themselves. This procedure is based on a modified ‘dressing’ method. For INEEs in 1+1 dimensions, some appropriate symmetry equations are shown to form the vector fields on a circle S1 algebra representation. In contrast to the so-called isospectral deformations, these symmetries result from conformal transformations of the associated linear problem spectrum. For INEEs in 2+1 dimensions, the commutation relations for symmetry equations are shown to coincide with operators \(\lambda ^m \partial _\lambda \), with integer m, p. Some additional results about Kac-Moody algebra applications are presented.