Periodic finite-band solutions to the focusing nonlinear Schrödinger equation by the Fokas method: inverse and direct problems

Abstract

We consider the Riemann–Hilbert (RH) approach to the construction of periodic finite-band solutions to the focusing nonlinear Schrödinger (NLS) equation. An RH problem for the solution of the finite-band problem has been recently derived via the Fokas method (Deconinck et al. 2021 Lett. Math. Phys. 111, 1–18. (doi:10.1007/s11005-021-01356-7); Fokas & Lenells. 2021 Proc. R. Soc. A 477, 20200605. (doi:10.1007/s11005-021-01356-7)) Building on this method, a finite-band solution to the NLS equation can be given in terms of the solution of an associated RH problem, the jump conditions for which are characterized by specifying the endpoints of the arcs defining the contour of the RH problem and the constants (so-called phases) involved in the jump matrices. In our work, we solve the problem of retrieving the phases given the solution of the NLS equation evaluated at a fixed time. Our findings are corroborated by numerical examples of phases computation, demonstrating the viability of the method proposed.

Publication DOI: https://doi.org/10.1098/rspa.2023.0828
Divisions: College of Engineering & Physical Sciences > Aston Institute of Photonics Technology (AIPT)
Aston University (General)
Funding Information: S. Bogdanov and J. E. Prilepsky acknowledge the support from Leverhulme Trust, grant no. RPG-2018-063.
Additional Information: Copyright © 2024, The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License https://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.
Uncontrolled Keywords: periodic finite-band solutions,Riemann–Hilbert problem,Fokas method,nonlinear Schrödinger equation
Publication ISSN: 1471-2946
Data Access Statement: The data and codes for the figures are available from the GitHub repository: https://github.com/Stepan0001/RHP-Direct-problem.git
Last Modified: 02 Dec 2024 09:05
Date Deposited: 04 Apr 2024 15:16
Full Text Link:
Related URLs: https://royalso ... /rspa.2023.0828 (Publisher URL)
https://github. ... ect-problem.git (Related URL)
http://www.scop ... tnerID=8YFLogxK (Scopus URL)
PURE Output Type: Article
Published Date: 2024-03-27
Published Online Date: 2024-03-27
Accepted Date: 2024-02-22
Authors: Shepelsky, Dmitry
Karpenko, Iryna
Bogdanov, Stepan
Prilepsky, Jaroslaw E. (ORCID Profile 0000-0002-3035-4112)

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