Integrable hierarchies with zero dispersion and elliptic curves

Abstract: We consider integrable hierarchies such as KP, modified KP, 2D Toda lattice, BKP (small and large), DKP, Pfaff-Toda and their multi-component generalizations. We work in the framework of the bilinear formalism in which the universal dependent variable is a tau-function satisfying bilinear equations of the Hirota-Miwa type. Our principal interest in this paper is the dispersionless versions of the hierarchies. In the limit of zero dispersion the main object is an $F$-function, which is the limit of properly re-scaled logarithm of the tau-function. We show that in all the cases there exists an algebraic curve built into the structure of the hierarchy. We call it the {\it dynamical curve}. For the KP, modified KP and Toda lattice hierarchies, as well as for their multi-component generalizations, the curve is rational (of genus 0) and can be uniformized by rational or trigonometric functions. For hierarchies of the Pfaff type (DKP and Pfaff-Toda) the dynamical curve is in general a smooth elliptic curve (of genus 1), with its modular parameter being a dynamical variable. It is also shown that the large BKP hierarchy admits two different dispersionless versions. In one of them the dynamical curve degenerates to a rational curve while in the other one it remains to be elliptic. We show that a reformulation of the hierarchies based on uniformization of the dynamical curves by elliptic (or trigonometric) functions makes their structure nice and clear, especially in the multi-component case.