Caudrelier, V orcid.org/0000-0003-0129-6758 and Crampé, N (2019) New integrable boundary conditions for the Ablowitz–Ladik model: From Hamiltonian formalism to nonlinear mirror image method. Nuclear Physics B, 946. 114720. ISSN 0550-3213
Abstract
Using Sklyanin's classical theory of integrable boundary conditions, we use the Hamiltonian approach to derive new integrable boundary conditions for the Ablowitz–Ladik model on the finite and half infinite lattice. In the case of half infinite lattice, the special and new emphasis of this paper is to connect directly the Hamiltonian approach, based on the classical r-matrix, with the zero curvature representation and Bäcklund transformation approach that allows one to implement a nonlinear mirror image method and construct explicit solutions. It is shown that for our boundary conditions, which generalise (discrete) Robin boundary conditions, a nontrivial extension of the known mirror image method to what we call time-dependent boundary conditions is needed. A careful discussion of this extension is given and is facilitated by introducing the notion of intrinsic and extrinsic picture for describing boundary conditions. This gives the specific link between Sklyanin's reflection matrices and Bäcklund transformations combined with folding, in the case of non-diagonal reflection matrices. All our results reproduce the known Robin boundary conditions setup as a special case: the diagonal case. Explicit formulas for constructing multisoliton solutions on the half-lattice with our time-dependent boundary conditions are given and some examples are plotted.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2019, The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. |
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: | 02 Aug 2019 11:49 |
Last Modified: | 25 Jun 2023 21:56 |
Status: | Published |
Publisher: | Elsevier BV |
Identification Number: | 10.1016/j.nuclphysb.2019.114720 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:149233 |