Hydrodynamic integrability via geometry

TL;DR

This paper introduces a geometric framework linking hydrodynamic integrability of quasilinear systems to compatible nets in the induced geometry, unifying previous results and extending to broader classes.

Contribution

It establishes a geometric criterion for hydrodynamic integrability based on compatible nets, generalizing known results through algebraic geometry and conjugate nets.

Findings

Hydrodynamic integrability is equivalent to the existence of compatible nets.

The approach unifies results across three subclasses of systems.

A generalized notion of conjugate nets is introduced.

Abstract

This paper develops a geometric approach to the theory of integrability by hydrodynamic reductions to establish an equivalence, for a large class of quasilinear systems, between hydrodynamic integrability and the existence of nets compatible with the geometry induced on the codomain of the system. This unifies and extends known results for three subclasses of such systems. The generalization is obtained by studying the algebraic geometry of the characteristic correspondence of the system, and by introducing a generalized notion of conjugate nets.