Compatible pairs of Hamiltonian operators of the first and third orders
Exactly Solvable and Integrable Systems
2026-02-17 v1
Abstract
We compute general compatibility conditions between a weakly nonlocal homogeneous Hamiltonian operator and a third-order homogeneous Hamiltonian operator. Such operators determine a bi-Hamiltonian structure for many integrable PDEs (Korteweg--De Vries, Camassa--Holm, dispersive water waves, Dym, WDVV and others). Remarkably, the full set of conditions is purely algebraic and the first-order operator is completely determined by commuting systems of conservation laws that are Hamiltonian with respect to a third-order operator. We illustrate the above results with several examples, some of which, concerning WDVV equations, are new.
Keywords
Cite
@article{arxiv.2602.14739,
title = {Compatible pairs of Hamiltonian operators of the first and third orders},
author = {Paolo Lorenzoni and Stanislav Opanasenko and Raffaele Vitolo},
journal= {arXiv preprint arXiv:2602.14739},
year = {2026}
}