Elliptic Families of Solutions of the Kadomtsev-Petviashvili Equation and the Field Elliptic Calogero-Moser System ∗

Abstract

We present a Lax pair for the field elliptic Calogero-Moser system and establish a connection between this system and the Kadomtsev-Petviashvili equation.Namely, we consider elliptic families of solutions of the KP equation such that their poles satisfy a constraint of being balanced.We show that the dynamics of these poles is described by a reduction of the field elliptic CM system. We construct a wide class of solutions to the field elliptic CM system by showing that any N -fold branched cover of an elliptic curve gives rise to an elliptic family of solutions of the KP equation with balanced poles. The main goal of this paper is to establish a connection between the field analog of the elliptic Calogero-Moser system (CM) introduced in (10) and the Kadomtsev-Petviashvili equation (KP). This connection is a next step along the line that goes back to the paper (1), where the dynamics of the poles of elliptic (rational or trigonometric) solutions of the Korteweg-de Vries equation (KdV) was described in terms of commuting flows of the elliptic (rational or trigonometric) CM system. It was shown in the earlier paper (7) of one of the authors that the constrained correspondence between a theory of the elliptic CM system and a theory of the elliptic solutions of the KdV equation becomes an isomorphism for the case of the KP equation.It turns out that a function u(x, y, t) that is an elliptic function with respect to the variable x satisfies the KP equation if and only if it has the form