Gauge Miura and Backlund Transformations for Generalized $A_n$-KdV Hierarchies

Abstract

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

arXiv commentLatex 18 pages, corrections and improvements in the text

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• An integrable hierarchy associated with loop extension of ${\mathbb{Z}}_2^2$-graded $\mathfrak{osp}(1|2)$
• New negative grade solitonic sector for supersymmetric KdV and mKdV hierarchies
• New soliton solutions for Chen-Lee-Liu and Burgers hierarchies and its Bäcklund transformations