Avan, J, Caudrelier, V orcid.org/0000-0003-0129-6758 and Crampé, N (2018) From Hamiltonian to zero curvature formulation for classical integrable boundary conditions. Journal of Physics A: Mathematical and Theoretical, 51 (30). 30LT01. ISSN 1751-8113
Abstract
We reconcile the Hamiltonian formalism and the zero curvature representation in the approach to integrable boundary conditions for a classical integrable system in 1 + 1 space-time dimensions. We start from an ultralocal Poisson algebra involving a Lax matrix and two (dynamical) boundary matrices. Sklyanin's formula for the double-row transfer matrix is used to derive Hamilton's equations of motion for both the Lax matrix and the boundary matrices in the form of zero curvature equations. A key ingredient of the method is a boundary version of the Semenov-Tian-Shansky formula for the generating function of the time-part of a Lax pair. The procedure is illustrated on the finite Toda chain for which we derive Lax pairs of size for previously known Hamiltonians of type BC N and D N corresponding to constant and dynamical boundary matrices respectively.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2018 IOP Publishing Ltd. This is an author-created, un-copyedited version of an article published in Journal of Physics A: Mathematical and Theoretical. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at https://doi.org/10.1088/1751-8121/aac976. |
Keywords: | integrable boundary conditions; Toda chain; classical r matrix; zero curvature representation |
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: | 04 Jun 2018 13:20 |
Last Modified: | 01 Jun 2019 00:39 |
Status: | Published |
Publisher: | IOP Publishing |
Identification Number: | 10.1088/1751-8121/aac976 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:131614 |