St\"{a}ckel representations of stationary KdV systems

TL;DR

This paper explores Stäckel representations of stationary KdV systems, demonstrating their dual separable forms, explicit transformations, and bi-Hamiltonian structure using Lax formalism.

Contribution

It establishes two distinct Stäckel representations of stationary KdV systems and constructs explicit transformations between jet coordinates and separation variables.

Findings

Identifies two different Stäckel representations of stationary KdV systems.

Provides explicit transformation between jet coordinates and separation variables.

Derives the bi-Hamiltonian structure via the Miura map.

Abstract

In this article we study St\"{a}ckel representations of stationary KdV systems. Using Lax formalism we prove that these systems have two different representations as separable St\"{a}ckel systems of Benenti type, related with different foliations of the stationary manifold. We do it by constructing an explicit transformation between the jet coordinates of stationary KdV systems and separation variables of the corresponding Benenti systems for arbitrary number of degrees of freedom. Moreover, on the stationary manifold, we present the explicit form of Miura map between both representations of stationary KdV systems, which also yields their bi-Hamiltonian formulation.