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

Dmitry Shepelsky*, Iryna Karpenko, Stepan Bogdanov, Jaroslaw E. Prilepsky

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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.
Original languageEnglish
Article number20230828
Number of pages28
JournalProceedings of the Royal Society of London A
Volume480
Issue number2286
Early online date27 Mar 2024
DOIs
Publication statusPublished - 27 Mar 2024

Bibliographical note

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.

Data Access Statement

The data and codes for the figures are available from the GitHub repository: https://github.com/Stepan0001/RHP-Direct-problem.git

Keywords

  • periodic finite-band solutions
  • Riemann–Hilbert problem
  • Fokas method
  • nonlinear Schrödinger equation

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