Geometric structures on the orbits of loop diffeomorphism groups and related heavenly-type Hamiltonian systems. I

Hentosh, O. E., Prykarpatskyy, Ya.A., Balinsky, Alexander ORCID: https://orcid.org/0000-0002-8151-4462 and Prykarpatski, A. K. 2023. Geometric structures on the orbits of loop diffeomorphism groups and related heavenly-type Hamiltonian systems. I. Ukrainian Mathematical Journal 10.1007/s11253-023-02129-2

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Abstract

We present a review of differential-geometric and Lie-algebraic approaches to the investigation of a broad class of nonlinear integrable differential systems of “heavenly” type associated with Hamiltonian flows on the spaces conjugate to the loop Lie algebras of vector fields on the tori. These flows are generated by the corresponding orbits of the coadjoint action of loop diffeomorphism groups and satisfy the Lax–Satotype vector-field compatibility conditions. We analyze the corresponding hierarchies of conservation laws and their relationships with Casimir invariants. We consider typical examples of these systems and establish their complete integrability by using the developed Lie-algebraic construction. We also describe new generalizations of the integrable dispersion-free systems of heavenly type for which the corresponding generating elements of the orbits have factorized structures, which allows their extension to the multidimensional case.

Item Type:	Article
Date Type:	Published Online
Status:	Published
Schools:	Mathematics
Publisher:	Springer
ISSN:	0041-5995
Date of First Compliant Deposit:	3 February 2023
Date of Acceptance:	1 July 2022
Last Modified:	03 Feb 2024 16:21
URI:	https://orca.cardiff.ac.uk/id/eprint/156491

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