Caudrelier, V orcid.org/0000-0003-0129-6758 and Stoppato, M orcid.org/0000-0002-2722-4931 (2021) Multiform description of the AKNS hierarchy and classical r-matrix. Journal of Physics A: Mathematical and Theoretical, 54 (23). 235204. ISSN 1751-8113
Abstract
In recent years, new properties of space-time duality in the Hamiltonian formalism of certain integrable classical field theories have been discovered and have led to their reformulation using ideas from covariant Hamiltonian field theory: in this sense, the covariant nature of their classical r-matrix structure was unravelled. Here, we solve the open question of extending these results to a whole hierarchy. We choose the Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy. To do so, we introduce for the first time a Lagrangian multiform for the entire AKNS hierarchy. We use it to construct explicitly the necessary objects introduced previously by us: a symplectic multiform, a multi-time Poisson bracket and a Hamiltonian multiform. Equipped with these, we prove the following results: (i) the Lax form containing the whole sequence of Lax matrices of the hierarchy possesses the rational classical r-matrix structure; (ii) the zero curvature equations of the AKNS hierarchy are multiform Hamilton equations associated to our Hamiltonian multiform and multi-time Poisson bracket; (iii) the Hamiltonian multiform provides a way to characterise the infinite set of conservation laws of the hierarchy reminiscent of the familiar criterion {I, H} = 0 for a first integral I.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2021 The Author(s). Published by IOP Publishing Ltd. Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. |
Keywords: | classicalr-matrix, integrable hierarchy, Lagrangian and Hamiltonian multiform |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 27 Apr 2021 10:59 |
Last Modified: | 25 Jun 2023 22:38 |
Status: | Published |
Publisher: | Institute of Physics Publishing |
Identification Number: | 10.1088/1751-8121/abfac9 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:173398 |
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