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.
We develop a systematic approach to deriving rational solutions and obtaining classification of their parameters for dressing chains of even N periodicity or equivalent Painlevé equations invariant under AN−1(1) symmetry. This formalism identifies rational solutions (as well as special function solutions) with points on orbits of fundamental shift operators of AN−1(1) affine Weyl groups acting on seed configurations defined as first-order polynomial solutions of the underlying dressing chains. This approach clarifies the structure of rational solutions and establishes an explicit and systematic method towards their construction. For the special case of the N=4 dressing chain equations, the method yields all the known rational (and special function) solutions of the Painlevé V equation. The formalism naturally extends to N=6 and beyond as shown in the paper.
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.
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