Integrable (2+1)-dimensional systems of hydrodynamic type

TL;DR

This paper reviews the classification of integrable (2+1)-dimensional hydrodynamic systems, highlighting Gibbons--Tsarev type systems and introducing new models related to algebraic curves of genus 2.

Contribution

It provides a comprehensive classification framework for these systems, including new models associated with higher-genus algebraic curves.

Findings

Known GT systems for genus 0 and 1 curves

Introduction of a new GT system for genus 2 curves

Construction of new integrable models from trivial GT systems

Abstract

We describe the results that have so far been obtained in the classification problem for integrable (2+1)-dimensional systems of hydrodynamic type. The systems of Gibbons--Tsarev type are the most fundamental here. A whole class of integrable (2+1)-dimensional models is related to each such system. We present the known GT systems related to algebraic curves of genus g=0 and g=1 and also a new GT system corresponding to algebraic curves of genus g=2. We construct a wide class of integrable models generated by the simplest GT system, which was not considered previously because it is in a sense trivial.