The elliptic lattice KdV system revisited

F. W. Nijhoff^a, C. Zhang^b, D.-J. Zhang<sup>b

a</sup> School of Mathematics, University of Leeds, Leeds, United Kingdom
^b Department of Mathematics, Shanghai University, Shanghai, China

Abstract: In our previous paper, a two-parameter extension of the lattice potential KdV equation was derived, associated with an elliptic curve. This comprises a rather complicated three-component system on the quad lattice, which contains the moduli of the elliptic curve as parameters. In this paper, we investigate this system further and, among other results, derive a two-component multiquartic form of the system on the quad lattice. Furthermore, we construct an elliptic Yang–Baxter map and study the associated continuous and semidiscrete systems. In particular, we derive the so-called “generating PDE” for this system, comprising a six-component system of second-order PDEs, which can be considered to constitute an elliptic extension of the Ernst equations of General Relativity.

Keywords: integrable quad-lattice equations, elliptic curves, Yang–Baxter maps, generating PDE.

Received: 13.02.2025
Revised: 13.02.2025

DOI: 10.4213/tmf10936

 English version: Theoretical and Mathematical Physics, 2025, 224:1, 1234–1256

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