Caudrelier, V (2018) Interplay between the Inverse Scattering Method and Fokas's Unified Transform with an Application. Studies in Applied Mathematics, 140 (1). pp. 3-26. ISSN 0022-2526
Abstract
It is known that the initial‐boundary value problem for certain integrable Partial Differential Equations (PDEs) on the half‐line with integrable boundary conditions can be mapped to a special case of the inverse scattering method (ISM) on the full‐line. This can also be established within the so‐called unified transform (UT) of Fokas for initial‐boundary value problems with linearizable boundary conditions. In this paper, we show a converse to this statement within the Ablowitz‐Kaup‐Newell‐Segur (AKNS) scheme: the ISM on the full‐line can be mapped to an initial‐boundary value problem with linearizable boundary conditions. To achieve this, we need a matrix version of the UT that was introduced by the author to study integrable PDEs on star‐graphs. As an application of the result, we show that the new, nonlocal reduction of the AKNS scheme introduced by Ablowitz and Musslimani to obtain the nonlocal nonlinear Schrödinger (NLS) equation can be recast as an old, local reduction, thus putting the nonlocal NLS and the NLS equations on equal footing from the point of view of the reduction group theory of Mikhailov.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2017 Wiley Periodicals, Inc., A Wiley Company. This is the peer reviewed version of the following article: Caudrelier, V. (2017), Interplay between the Inverse Scattering Method and Fokas's Unified Transform with an Application. Studies in Applied Mathematics. doi: 10.1111/sapm.12190, which has been published in final form at https://doi.org/10.1111/sapm.12190. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. |
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 Sep 2017 10:39 |
Last Modified: | 31 Aug 2018 00:39 |
Status: | Published |
Publisher: | Wiley |
Identification Number: | 10.1111/sapm.12190 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:121682 |