Nonlinear mirror image method for nonlinear Schrödinger equation: Absorption/emission of one soliton by a boundary

Abstract

We perform the analysis of the focusing nonlinear Schrödinger equation on the half-line with time-dependent boundary conditions along the lines of the nonlinear method of images with the help of Bäcklund transformations . The difficulty arising from having such time-dependent boundary conditions at x=0 is overcome by changing the viewpoint of the method and fixing the Bäcklund transformation at infinity as well as relating its value at x=0 to a time-dependent reflection matrix. The interplay between the various aspects of integrable boundary conditions is reviewed in detail to brush a picture of the area. We find two possible classes of solutions. One is very similar to the case of Robin boundary conditions whereby solitons are reflected at the boundary, as a result of an effective interaction with their images on the other half-line . The new regime of solutions supports the existence of one soliton that is not reflected at the boundary but can be either absorbed or emitted by it. We demonstrate that this is a unique feature of time-dependent integrable boundary conditions.

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Item Type:	Article
Authors/Creators:	Caudrelier, V https://orcid.org/0000-0003-0129-6758 Crampé, N Dibaya, CM https://orcid.org/0000-0001-9838-8362
Copyright, Publisher and Additional Information:	© 2021 Wiley Periodicals LLC. This is the peer reviewed version of the following article: Caudrelier, V, Crampé, N, Dibaya, CM. Nonlinear mirror image method for nonlinear Schrödinger equation: Absorption/emission of one soliton by a boundary. Stud Appl Math. 2021; 1– 43. https://doi.org/10.1111/sapm.12456 , which has been published in final form at https://doi.org/10.1111/sapm.12456. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. This article may not be enhanced, enriched or otherwise transformed into a derivative work, without express permission from Wiley or by statutory rights under applicable legislation. Copyright notices must not be removed, obscured or modified. The article must be linked to Wiley’s version of record on Wiley Online Library and any embedding, framing or otherwise making available the article or pages thereof by third parties from platforms, services and websites other than Wiley Online Library must be prohibited.
Keywords:	integrable initial-boundary value problem, inverse scatteringmethod, nonlinear Schrödinger equation
Dates:	Published (online): 18 October 2021 Accepted: 17 September 2021
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:	13 Oct 2021 13:45
Last Modified:	18 Oct 2022 00:25
Status:	Published online
Publisher:	Wiley
Identification Number:	10.1111/sapm.12456
Open Archives Initiative ID (OAI ID):	oai:eprints.whiterose.ac.uk:179087

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Nonlinear mirror image method for nonlinear Schrödinger equation: Absorption/emission of one soliton by a boundary

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