Scaling Symmetry Reductions of Coupled KdV Systems

Allan P Fordy

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arXiv · 2405.09694

Scaling Symmetry Reductions of Coupled KdV Systems

Allan P Fordy

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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.

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Allan P Fordy. 2024-05-15. Scaling Symmetry Reductions of Coupled KdV Systems. https://arxiv.org/abs/2405.09694

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