Bihamiltonian tests for integrable systems associated to rank-$1$ F-CohFTs

TL;DR

This paper investigates the bihamiltonian properties of DR hierarchies linked to rank-1 F-CohFTs, revealing connections to well-known integrable hierarchies like Camassa-Holm and Degasperis-Procesi.

Contribution

It performs bihamiltonian tests on these hierarchies, conjectures a parameter family with Miura equivalence to Camassa-Holm, and explores relations to Degasperis-Procesi hierarchies.

Findings

Bihamiltonian tests support a 2-parameter family of hierarchies.

A 1-parameter subfamily is conjecturally Miura equivalent to Camassa-Holm.

Another subfamily is conjecturally related to Degasperis-Procesi hierarchy.

Abstract

Double ramification (DR) hierarchies associated to rank-$1$ F-CohFTs are important integrable perturbations of the Riemann--Hopf hierarchy. In this paper, we perform bihamiltonian tests for these DR hierarchies, and conjecture that the ones that are bihamiltonian form a $2$-parameter family. Remarkably, our computations suggest that there is a $1$-parameter subfamily of the rank-$1$ F-CohFTs, where the corresponding DR hierarchy is conjecturally Miura equivalent to the Camassa--Holm hierarchy. We also prove a conjecture regarding bihamiltonian Hodge hierarchies. Finally, we systematically study Miura invariants, and for another $1$-parameter subfamily propose a conjectural relation to the Degasperis--Procesi hierarchy.