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arXiv · 2602.14739

Compatible pairs of Hamiltonian operators of the first and third orders

Paolo Lorenzoni·Stanislav Opanasenko·Raffaele Vitolo
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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.

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Paolo Lorenzoni, Stanislav Opanasenko, Raffaele Vitolo. 2026-02-16. Compatible pairs of Hamiltonian operators of the first and third orders. https://doi.org/10.1088/1751-8121/ae751d

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