The elliptic lattice KdV system revisited

Abstract

In a previous paper [Nijhoff,Puttock,2003], a 2-parameter extension of the lattice potential KdV equation was derived, associated with an elliptic curve. This comprises a rather complicated 3-component system on the quad lattice which contains the moduli of the elliptic curve as parameters. In the present paper, we investigate this system further and, among other results, we derive a 2-component multiquartic form of the system on the quad lattice. Furthermore, we construct an elliptic Yang-Baxter map, and study the associated continuous and semi-discrete systems. In particular, we derive the so-called ``generating PDE'' for this system, comprising a 6-component system of second order PDEs which could be considered to constitute an elliptic extension of the Ernst equations of General Relativity.

25 pages, 1 diagram, submitted for the Proceedings of the 5th International Conference on Integrable Systems Nonlinear Dynamics, Yaroslavl, October 7-11, 2024