The generalised hodograph method for non-diagonalisable integrable systems of hydrodynamic type

TL;DR

This paper extends the generalized hodograph method to non-diagonalisable integrable hydrodynamic systems, enabling the construction of solutions based on symmetries and exploring conditions for its applicability.

Contribution

It introduces a generalized hodograph method for regular non-diagonalisable integrable systems, linking them to F-manifolds and detailing the conditions for solution construction.

Findings

The method applies to non-diagonalisable systems under specific assumptions.

Solutions can be constructed from symmetries similar to the diagonal case.

The approach generalizes Tsarev's diagonal solution method.

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.