Gauge Miura and Bäcklund transformations for generalized An-KdV hierarchies

Abstract

The construction of Miura and Bäcklund transformations for An mKdV and KdV hierarchies are presented in terms of gauge transformations acting upon the zero curvature representation. As in thewell knownsl(2) case, we derive and relate the equations of motion for the two hierarchies. Moreover, the Miuragauge transformation is not unique, instead, it is shown to be connected to a set of generators labeled by the exponents of An. The construction of generalized gauge-Bäcklund transformation for the An-KdV hierarchy is obtained as a composition ofMiura and Bäcklund-gauge transformations for An-mKdV hierarchy. The zero curvature representation provide a framework which is universal within all flows and generate systematically Bäcklund transformations for the entirely hierarchy.