Dispersionless Hirota equations and the genus 3 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.

arXiv commentamended version, to appear in Comm. Math. Phys., 15 pages

References in corpus (6)

• On integrability of (2+1)-dimensional quasilinear systems
• Conformal maps and dispersionless integrable hierarchies
• Einstein-Weyl geometry, the dKP equation and twistor theory
• Integrable Structure of the Dirichlet Boundary Problem in Two Dimensions
• On the integrability of symplectic Monge-Ampére equations
• Integrable Structure of the Dirichlet Boundary Problem in Multiply-Connected Domains

Cited by in corpus (3)

• Characteristic Lie Algebras of Integrable Differential-Difference Equations in 3D
• Second-order PDEs in 3D with Einstein-Weyl conformal structure
• Integrable partial differential equations and Lie--Rinehart algebras