The generalised hodograph method for non-diagonalisable integrable systems of hydrodynamic type
Paolo Lorenzoni, Sara Perletti, Karoline van Gemst
TL;DR
The paper addresses extending the generalised hodograph method to regular non-diagonalisable hydrodynamic-type systems by exploiting the geometry of F-manifolds with compatible connection. It develops a symmetry-based framework for constructing implicit solutions, generalising Tsarev’s diagonalisable approach to block-Toeplitz systems and identifying the role of Darboux-type completeness. The authors establish necessary and sufficient conditions for the completeness of symmetries in one-block and multi-block Jordan structures, demonstrating that the general solution depends on $n$ arbitrary functions of a single variable under appropriate restrictions, and illustrate the construction with Lauricella bi-flat F-manifolds. The work provides a rigorous geometric underpinning for integrating non-diagonalisable systems and connects integrability to Darboux-type theorems, with implications for Hamiltonian structures and future extensions.
Abstract
We extend the generalised hodograph method to regular non- diagonalisable integrable systems of hydrodynamic type, in light of the relation between such systems and F-manifolds with compatible connection. The method allows the construction of solutions starting from the symmetries of the system. In the diagonal case, the completeness of the symmetries follows from the integrability conditions that ensure the applicability of a Darboux’s theorem on Pfaffian systems. In the regular non-diagonalisable case the validity of this theorem relies on some further assumptions that we discuss in detail. Under these assumptions, the method provides the general solution as in Tsarev’s diagonal case.