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Caudrelier, V orcid.org/0000-0003-0129-6758 and Stoppato, M (2020) A connection between the classical r-matrix formalism and covariant Hamiltonian field theory. Journal of Geometry and Physics, 148. 103546. ISSN 0393-0440
Abstract
We bring together aspects of covariant Hamiltonian field theory and of classical integrable field theories in 1+1 dimensions. Specifically, our main result is to obtain for the first time the classical -matrix structure within a covariant Poisson bracket for the Lax connection, or Lax one form. This exhibits a certain covariant nature of the classical -matrix with respect to the underlying spacetime variables. The main result is established by means of several prototypical examples of integrable field theories, all equipped with a Zakharov–Shabat type Lax pair. Full details are presented for: (a) the sine–Gordon model which provides a relativistic example associated to a classical r-matrix of trigonometric type; (b) the nonlinear Schrödinger equation and the (complex) modified Korteweg–de Vries equation which provide two non-relativistic examples associated to the same classical r-matrix of rational type, characteristic of the AKNS hierarchy. The appearance of the r-matrix in a covariant Poisson bracket is a signature of the integrability of the field theory in a way that puts the independent variables on equal footing. This is in sharp contrast with the single-time Hamiltonian evolution context usually associated to the r-matrix formalism.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2019 Elsevier B.V. All rights reserved. This is an author produced version of a paper published in Journal of Geometry and Physics. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Classical r-matrix; Covariant Hamiltonian field theory; Lax connection; Multisymplectic form |
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: | 26 Nov 2019 13:49 |
Last Modified: | 02 Apr 2025 10:29 |
Status: | Published |
Publisher: | Elsevier |
Identification Number: | 10.1016/j.geomphys.2019.103546 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:153868 |
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A connection between the classical r-matrix formalism and covariant Hamiltonian field theory. (deposited 02 Apr 2025 10:28)
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