Poisson structures of some heavenly type dynamical systems

2023/12/09 by Prykarpatskyy, Yarema
#17B68 #17B80 #34A34 #35G25 #35N10 #35Q53 #37K05 #37K10 #37K35 #58J70 #58J72 #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2312.05618

Abstract

The paper investigates the Poisson structures associated with dynamical systems of the heavenly type, focusing on the Mikhalev-Pavlov and Plebański equation. The dynamical system is represented as a Hamiltonian system on a functional manifold, and Poisson brackets are defined based on a non-degenerate Poisson operator. The study explores the Lax-type integrability and bi-Hamiltonian properties of the systems, revealing the existence of compatible Poisson operators. The Lie-algebraic approach, particularly the AKS-algebraic and R-structure schemes, is employed to analyze the holomorphic loop Lie algebra, providing insights into the Lie-algebraic structure of heavenly equations. The Mikhalev-Pavlov and Plebański equations are studied in detail, and the associated Poisson brackets for specific coordinate functions are derived, revealing interesting mathematical properties. The paper establishes a foundation for understanding the symplectic structures associated with heavenly-type dynamical systems.

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