Hamiltonian formalism for non-diagonalisable systems of hydrodynamic type
Paolo Lorenzoni, Sara Perletti, Karoline van Gemst
TL;DR
The work extends Tsarev’s integrability framework from diagonal to non-diagonalisable hydrodynamic-type systems by embedding block-diagonal velocity structures into regular F-manifolds with compatible connection. It shows that, under Darboux–Tsarev conditions, the Dubrovin–Novikov–Tsarev system for the local Hamiltonian metric g becomes solvable and compatible, with the general solution governed by n arbitrary functions of a single variable (and additional two-variable functions per block). The approach leverages canonical (David–Hertling) coordinates, a torsionless ∇ compatible with the product, and a generalized hodograph method to produce solutions for both symmetry and conservation-law densities. This provides a non-diagonalisable analogue of Tsarev’s semi-Hamiltonian theory, clarifying when Hamiltonian structures exist and how many functional degrees of freedom govern the metric in block-structured 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.