Characterization of DR hierarchies

Determine whether a given hierarchy of evolutionary partial differential equations belongs to the class of double ramification hierarchies.

Background

The paper discusses Hamiltonian and quantum double ramification hierarchies constructed from cohomological field theories. Although many known integrable hierarchies arise as double ramification hierarchies, the paper states that no general criterion is currently known for deciding whether an arbitrary hierarchy of evolutionary partial differential equations has this origin. Several conjectures are known only in the one-dependent-variable case.

References

Note, however, that it is still not known how to determine whether a given hierarchy of evolutionary PDEs belongs to the class of DR hierarchies or not, although there are several conjectures in the case when there is only one dependent variable.

— Quantum hierarchies of the ILW and BO equations and the DR cycles on the moduli spaces of curves  (2610.01841 - Buryak et al., 1 Oct 2026) in Section 1, Introduction