Hamiltonian formalism for non-diagonalisable systems of hydrodynamic type

TL;DR

This paper extends the Hamiltonian formalism for hydrodynamic systems to non-diagonalisable cases, exploring integrability conditions and solution existence beyond the classical diagonal framework.

Contribution

It generalizes the Hamiltonian formalism and integrability conditions to non-diagonalisable hydrodynamic systems, broadening the scope of classical results.

Findings

Established conditions for integrability in non-diagonalisable systems

Proved existence of solution families depending on functional parameters

Extended classical theorems to more general hydrodynamic systems

Abstract

We study the system of first order PDEs for pseudo-Riemannian metrics governing the Hamiltonian formalism for systems of hydrodynamic type. In the diagonal setting the integrability conditions ensure the compatibility of this system and, thanks to a classical theorem of Darboux, the existence of a family of solutions depending on functional parameters. In this paper we study the generalisation of this result to a class of non-diagonalisable systems of hydrodynamic type that naturally generalises Tsarev's integrable diagonal systems.