On the status of DELL systems
A. Mironov, A. Morozov
TL;DR
This paper reviews the duality properties of Calogero systems, introduces new elliptic-trigonometric Hamiltonians, and discusses the open problem of the self-dual double-elliptic (DELL) system.
Contribution
It presents the first elliptic-trigonometric Hamiltonians dual to elliptic Ruijsenaars Hamiltonians and clarifies the formulation of the open DELL system problem.
Findings
Introduction of elliptic-trigonometric dual Hamiltonians
Relations between these Hamiltonians and the bi-elliptic KS model
Outline of steps to address the DELL system mystery
Abstract
A detailed review of the -duality for Calogero system and its generalizations is given. For the first time, we present some of elliptic-trigonometric Hamiltonians dual to the elliptic Ruijsenaars Hamiltonians (i.e. trigonometric-elliptic ones), and explain their relations to the bi-elliptic Koroteev-Shakirov (KS) model. The most interesting self-dual double-elliptic (DELL) system remains a mystery, but we provide a clearer formulation of the problem and describe the steps that are still to be done.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Nonlinear Waves and Solitons · Advanced Topics in Algebra