Gauge transformations and symmetries of integrable systems

Abstract

We analyse several integrable systems in zero-curvature form within the framework of invariant gauge theory. In the Drinfeld–Sokolov gauge, we derive a two-parameter family of nonlinear evolution equations which as special cases include the Korteweg–de Vries (KdV) and Harry Dym (HD) equations. We find residual gauge transformations which lead to infinitesimal symmetries of this family of equations. For KdV and HD equations we find an infinite hierarchy of such symmetry transformations, and we investigate their relation with local conservation laws, constants of the motion and the bi-Hamiltonian structure of the equations. Applying successive gauge transformations of Miura type we obtain a sequence of gauge-equivalent integrable systems, among them the modified KdV and Calogero KdV equations.