Integrable structures of dispersionless systems and differential geometry

TL;DR

This paper develops a differential-geometric framework for integrable dispersionless systems, constructs associated Gibbons-Tsarev systems for algebraic curves, and proves the integrability of the universal Whitham hierarchy.

Contribution

It introduces a geometric approach to Whitham hierarchies and establishes their integrability via hydrodynamic reductions, connecting algebraic geometry with integrable systems.

Findings

Construction of Gibbons-Tsarev systems for arbitrary genus algebraic curves

Proof of integrability of the universal Whitham hierarchy

Development of a differential-geometric theory for dispersionless integrable systems

Abstract

We develop the theory of Whitham type hierarchies integrable by hydrodynamic reductions as a theory of certain differential-geometric objects. As an application we construct Gibbons-Tsarev systems associated to moduli space of algebraic curves of arbitrary genus and prove that the universal Whitham hierarchy is integrable by hydrodynamic reductions.