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.
Related
- Elliptic orbifold lines and integrable hierarchies
- THE ZERO-DISPERSION LIMIT OF THE PERIODIC BENJAMIN-ONO EQUATION FOR SQUARE-INTEGRABLE DATA
- On Sufficient Conditions for Rank Zero in Families of Elliptic Curves
- Elliptic spin Ruijsenaars-Schneider integrable models from 5d 𝒩=1 gauge theories
- Cetraro lectures on integrable systems and moduli spaces of curves
BibTeX
@misc{savchenko2026integrable,
title = {{Integrable hierarchies with zero dispersion and elliptic curves}},
author = {A. Savchenko and A. Zabrodin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.01354},
eprint = {2606.01354},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.01354},
}
RIS
TY - UNPB
TI - Integrable hierarchies with zero dispersion and elliptic curves
AU - A. Savchenko
AU - A. Zabrodin
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2606.01354
UR - https://arxiv.org/abs/2606.01354
AB - 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.
ER -