Classification of integrable two-component Hamiltonian systems of hydrodynamic type in 2+1 dimensions

TL;DR

This paper classifies all integrable two-component Hamiltonian systems of hydrodynamic type in 2+1 dimensions, revealing their structure through elliptic and hypergeometric functions and advancing the understanding of higher-dimensional integrability.

Contribution

It provides a complete classification of integrable two-component Hamiltonian systems in 2+1 dimensions using differential-geometric Poisson brackets and hypergeometric functions.

Findings

Classification of integrable systems in 2+1 dimensions.

Parametrization of Hamiltonians via elliptic/hypergeometric functions.

Introduction of a novel hypergeometric-based Godunov system construction.

Abstract

Hamiltonian systems of hydrodynamic type occur in a wide range of applications including fluid dynamics, the Whitham averaging procedure and the theory of Frobenius manifolds. In 1+1 dimensions, the requirement of the integrability of such systems by the generalised hodograph transform implies that integrable Hamiltonians depend on a certain number of arbitrary functions of two variables. On the contrary, in 2+1 dimensions the requirement of the integrability by the method of hydrodynamic reductions, which is a natural analogue of the generalised hodograph transform in higher dimensions, leads to finite-dimensional moduli spaces of integrable Hamiltonians. In this paper we classify integrable two-component Hamiltonian systems of hydrodynamic type for all existing classes of differential-geometric Poisson brackets in 2D, establishing a parametrisation of integrable Hamiltonians via elliptic/hypergeometric functions. Our approach is based on the Godunov-type representation of Hamiltonian systems, and utilises a novel construction of Godunov's systems in terms of generalised hypergeometric functions.