The elliptic lattice KdV system revisited
Frank Nijhoff, Cheng Zhang, Da-jun Zhang
TL;DR
The paper advances the theory of elliptic lattice KdV systems by revisiting EllΔΔKdV, a two-parameter elliptic extension of lattice KdV on a quad lattice with elliptic moduli. It develops a comprehensive algebraic framework based on elliptic matrices and a Cauchy kernel to construct Lax pairs, a tau-function, and a six-component generating PDE that extends the Ernst equations of general relativity. It also derives a 2-component multiquartic form, an elliptic Yang-Baxter map, and semi-discrete and continuous reductions, including an elliptic Toda type system and an elliptic dressing chain, enriching the integrable structure with elliptic geometry. The results show MDC and provide explicit soliton-type Cauchy–determinant solutions, while highlighting open questions such as r-matrix formulations, algebro-geometric solutions, and deeper connections to KP-type hierarchies and elliptic Ernst dynamics, signaling broad implications for elliptic discrete integrable systems.
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.