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Dispersionless Hirota equations and the genus 3 hyperelliptic divisor

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journal contribution
posted on 28.08.2019, 09:52 by Fabien Clery, Evgeny FerapontovEvgeny Ferapontov
Equations of dispersionless Hirota type F(uxixj ) = 0 have been thoroughly investigated in mathematical physics and differential geometry. It is known that the parameter space of integrable Hirota type equations in 3D is 21-dimensional, and that the action of the natural equivalence group Sp(6, R) on the parameter space has an open orbit. However the structure of the generic equation corresponding to the open orbit remained elusive. Here we prove that the generic 3D Hirota equation is given by the remarkable formula ϑm(τ ) = 0, τ = i Hess(u) where ϑm is any genus 3 theta constant with even characteristics and Hess(u) is the 3 × 3 Hessian matrix of a (real-valued) function u(x1, x2, x3). Thus, generic Hirota equation coincides with the equation of the genus 3 hyperelliptic divisor (to be precise, its intersection with the imaginary part of the Siegel upper half space H3). The rich geometry of integrable Hirota type equations sheds new light on local differential geometry of the genus 3 hyperelliptic divisor, in particular, the integrability conditions can be viewed as local differential-geometric constraints that characterise the hyperelliptic divisor uniquely modulo Sp(6, C)-equivalence.

Funding

This research was supported by the EPSRC grant EP/N031369/1.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Communications in Mathematical Physics

Volume

376

Issue

2

Pages

1397 - 1412

Citation

CLERY, F. and FERAPONTOV, E.V., 2020. Dispersionless Hirota equations and the genus 3 hyperelliptic divisor. Communications in Mathematical Physics, 376 (2), pp.1397-1412.

Publisher

Springer

Version

VoR (Version of Record)

Rights holder

© The Author(s)

Publisher statement

This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Acceptance date

08/07/2019

Publication date

2019-08-19

Copyright date

2019

ISSN

0010-3616

eISSN

1432-0916

Language

en