The elliptic lattice KdV system revisited

TL;DR

This paper revisits an elliptic lattice KdV system, deriving a multiquartic form, constructing an elliptic Yang-Baxter map, and exploring related continuous and semi-discrete systems, including a generating PDE akin to elliptic Ernst equations.

Contribution

It introduces a new multiquartic form of the elliptic lattice KdV system and constructs an elliptic Yang-Baxter map, expanding understanding of elliptic integrable systems.

Findings

Derived a 2-component multiquartic form of the system

Constructed an elliptic Yang-Baxter map

Formulated a generating PDE similar to elliptic Ernst equations

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.