Periodic ILW equation with discrete Laplacian

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.