Scaling Symmetry Reductions of Coupled KdV Systems

TL;DR

This paper explores Painlevé reductions of coupled KdV systems, revealing structural similarities with stationary reductions and deriving compatible Poisson brackets and a discrete Darboux transformation.

Contribution

It introduces a unified approach to Painlevé reductions for coupled KdV systems and derives new compatible Poisson brackets and a discrete Darboux transformation.

Findings

Derived 2 and 3 compatible Poisson brackets with stationary structure.

Established a discrete Darboux transformation as a non-autonomous generalization.

Showed the parallelism between Painlevé and stationary reductions.

Abstract

In this paper we discuss the Painlev\'e reductions of coupled KdV systems. We start by comparing the procedure with that of {\em stationary reductions}. Indeed, we see that exactly the same construction can be used at each step and parallel results obtained. For simplicity, we restrict attention to the $t_2$ flow of the KdV and DWW hierarchies and derive respectively 2 and 3 compatible Poisson brackets, which have identical {\em structure} to those of their stationary counterparts. In the KdV case, we derive a discrete version, which is a non-autonomous generalisation of the well known Darboux transformation of the stationary case.