Classification of bi-Hamiltonian pairs extended by isometries
Maxim V. Pavlov, Pierandrea Vergallo, Raffaele Vitolo
TL;DR
This paper classifies bi-Hamiltonian pairs of Dubrovin-Novikov type with isometric non-local parts, providing new examples and extending known systems, advancing understanding of integrable Hamiltonian structures.
Contribution
It offers a classification of bi-Hamiltonian pairs with isometric non-local parts for two variables, including a new example extending hydrodynamic systems.
Findings
Classification of bi-Hamiltonian pairs with isometric non-local parts
Identification of a new extended hydrodynamic type system
Examples related to the constant astigmatism equation
Abstract
The aim of this article is to classify pairs of first-order Hamiltonian operators of Dubrovin-Novikov type such that one of them has a non-local part defined by an isometry of its leading coefficient. An example of such bi-Hamiltonian pair was recently found for the constant astigmatism equation. We obtain a classification in the case of 2 dependent variables, and a significant new example that is an extension of a hydrodynamic type system obtained from a particular solution of the WDVV equations.
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