Compatible pairs of Hamiltonian operators of the first and third orders
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910023777517568 |
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| author | Lorenzoni, Paolo Opanasenko, Stanislav Vitolo, Raffaele |
| author_facet | Lorenzoni, Paolo Opanasenko, Stanislav Vitolo, Raffaele |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14739 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Compatible pairs of Hamiltonian operators of the first and third orders Lorenzoni, Paolo Opanasenko, Stanislav Vitolo, Raffaele Exactly Solvable and Integrable Systems 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. |
| title | Compatible pairs of Hamiltonian operators of the first and third orders |
| topic | Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2602.14739 |