Dispersionless Hirota equations and the genus 3 hyperelliptic divisor

TL;DR

This paper identifies the master-equation for dispersionless Hirota equations in 3D as the vanishing of genus 3 theta constants, linking integrability conditions to the geometry of the genus 3 hyperelliptic divisor.

Contribution

It proves that the master-equation corresponds to genus 3 theta constant vanishing, revealing the geometric structure of integrable Hirota equations in 3D.

Findings

Master-equation characterized by genus 3 theta constant vanishing

Connection between integrability conditions and hyperelliptic divisor geometry

Local differential geometric constraints uniquely define the hyperelliptic divisor

Abstract

Equations of dispersionless Hirota type have been thoroughly investigated in the mathematical physics and differential geometry literature. It is known that the parameter space of integrable Hirota type equations in 3D is 21-dimensional and the action of the natural equivalence group Sp(6, R) on the parameter space has an open orbit. However the structure of the `master-equation' corresponding to this orbit remained elusive. Here we prove that the master-equation is specified by the vanishing of any genus 3 theta constant with even characteristic. 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.