Generalized Bäcklund transformations for affine Toda 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.

Gauge Miura and Bäcklund transformations for generalized A n -KdV hierarchies

“…A natural construction of Bäcklund-gauge transformations U(φ i , ψ i ) in terms of a graded affine algebra was proposed in [7,8] for the A 1 mKdV (sinh-Gordon) and in [9] for the generalized A n Toda hierarchies. Moreover U(φ i , ψ i ) was shown to be an universal object and provide a systematic construction of Bäcklund transformation extended to all evolution equations (flows) within the hierarchy since (1.2) is valid for all flows (i.e.…”

“…In this section we generalize the results of reference [8] by constructing the Bäcklund transformation for the A n -KdV hierarchy from the Miura and Bäcklund-gauge transformations constructed for the A n -mKdV hierarchy. Consider the Bäcklund-gauge transformation for the A n -mkdV hierarchy proposed in [9],…”

“…Here we will follow the same principle by choosing different S solutions for right and left multiplication in (6.4). Bäcklund transformation for the A n Toda theory was first proposed in [20] and in [9], it was generalized as a gauge transformation constructed and classified according to a graded affine Lie algebraic structure. Such construction was shown to be universal within the hierarchy, i.e.…”

“…Consider, as an illustration, the type I Bäcklund-gauge transformation for sl(3) mKdV hierarchy (see for instance [9]),…”

“…Soliton solutions in turn, can be constructed by gauge transforming a vacuum solution A vac x and A vac t into some nontrivial configuration (dressing) see for instance [5]. More recently, gauge transformation was shown to be a key ingredient in constructing Bäcklund transformations [6][7][8][9]. In all cases, gauge transformation is an essential ingredient.…”