SKdV, SmKdV flows and their supersymmetric gauge-Miura transformations

The construction of Integrable Hierarchies in terms of zero curvature representation provides a systematic construction for a series of integrable non-linear evolution equations (flows) which shares a common affine Lie algebraic structure. The integrable hierarchies are then classified in terms of a decomposition of the underlying affine Lie algebra Gˆ into graded subspaces defined by a grading operator Q. In this paper we shall discuss explicitly the simplest case of the affine şl(2) Kac-Moody algebra within the principal gradation given rise to the KdV and mKdV hierarchies and extend to supersymmetric models. Inspired by the dressing transformation method, we have constructed a gaugeMiura transformation mapping mKdV into KdV flows. Interesting new results concerns the negative grade sector of the mKdV hierarchy in which a double degeneracy of flows (odd and its consecutive even) of mKdV are mapped into a single odd KdV flow. These results are extended to supersymmetric hierarchies based upon the affine şl(2, 1) super-algebra.