Gibbons-Tsarev type systems and Eventual identities

TL;DR

This paper investigates the structure of Gibbons-Tsarev systems related to non-semisimple F-manifolds, proving non-existence of certain reductions and constructing integrable reductions of Pavlov's chain.

Contribution

It introduces a generalized Gibbons--Tsarev system for non-semisimple cases and links solutions to eventual identities of F-manifolds, advancing integrable systems theory.

Findings

Non-diagonalisable reductions of the dKP equation cannot exist.

A class of solutions is characterized by eventual identities.

Constructed integrable reductions of Pavlov's hydrodynamic chain.

Abstract

We show that non-diagonalisable reductions of the dKP equation associated with regular non-semisimple $F$-manifolds cannot exist. The proof is based on the derivation and study of a generalised Gibbons--Tsarev system (gGT system) in the non-semisimple/non-diagonalisable setting. Remarkably, a class of solutions of the gGT system is defined by eventual identities of the underlying regular $F$-manifold structure. Furthermore, we use these vector fields to construct integrable reductions of Pavlov's hydrodynamic chain. In this case, the corresponding solutions are defined for any choice of Jordan block structure of the operator of multiplication by an eventual identity.