Periodic ILW equation with discrete Laplacian

TL;DR

This paper introduces a generalized periodic ILW equation with a discrete Laplacian, demonstrating its integrability, constructing special solutions including elliptic ones, and exploring its algebraic structures and conjectured hierarchies.

Contribution

It extends the periodic ILW equation by incorporating a discrete Laplacian, establishes its integrability, and connects it to elliptic functions, Poisson structures, and quantum algebras.

Findings

Proves integrability of the generalized ILW equation.

Constructs special solutions, including elliptic solutions.

Proposes a conjecture linking the hierarchy to a Poisson algebra and quantum algebra.

Abstract

We study an integro-differential equation which generalizes the periodic intermediate long wave (ILW) equation. The kernel of the singular integral involved is an elliptic function written as a second order difference of the Weierstrass zeta-function. Using Sato's formulation, we show the integrability and construct some special solutions. An elliptic solution is also obtained. We present a conjecture based on a Poisson structure that it gives an alternative description of this integrable hierarchy. We note that this Poisson algebra in turn is related to a quantum algebra related with the family of Macdonald difference operators.