Local bi-integrability of bi-Hamiltonian systems via bi-Poisson reduction

2024/10/27 by I. K. Kozlov, Kozlov, I. K.
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Waves and Solitons #Numerical methods for differential equations #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2410.20574

openalex publication_date 2024/10/27 · openalex created_date 2024/11/14 · openalex updated_date 2026/07/28

Abstract

We prove that any bi-Hamiltonian system v = (A + λB)dH_λ that is Hamiltonian with respect all Poisson brackets A + λB is locally bi-integrable in both the real smooth case, when all eigenvalues of the Poisson pencil P = \A + λB\ are real, and in the complex analytic case. A complete set of functions in bi-involution is constructed by extending the set of standard integrals, which consists of Casimir functions of Poisson brackets, eigenvalues of the Poisson pencil and Hamiltonians.

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