Stationary coupled KdV systems and their St\"ackel representations

TL;DR

This paper demonstrates that stationary coupled KdV systems can be reformulated as classical separable St"ackel systems in multiple ways, revealing new explicit mappings and a unified Hamiltonian structure via Miura maps.

Contribution

It establishes that stationary cKdV systems are equivalent to multiple St"ackel systems and introduces explicit maps and a finite-dimensional Miura transformation linking different representations.

Findings

Each stationary cKdV system admits (N+1) distinct St"ackel representations.

Explicit maps between jet variables and separation variables are constructed.

Different St"ackel representations are connected by finite-dimensional Miura maps.

Abstract

In this article we investigate stationary cKdV systems and prove that every $N$-field stationary cKdV system can be written, after a careful reparametrization of jet variables, as a classical separable St\"ackel system on $N+1$ different ways. For each of these $N+1$ parametrizations we present an explicit map between the jet variables and the separation variables of the system. Finally, we show that each pair of St\"ackel representations of the same stationary cKdV system, when considered in the phase space extended by Casimir variables, is connected by an appropriate finite-dimensional Miura map, which leads to an $(N+1)$-Hamiltonian formulation for the stationary cKdV system.