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

doi:10.1098/rspa.2023.0828

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 journal › Article › peer-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 language	English
Article number	20230828
Number of pages	28
Journal	Proceedings of the Royal Society of London A
Volume	480
Issue number	2286
Early online date	27 Mar 2024
DOIs	https://doi.org/10.1098/rspa.2023.0828
Publication status	Published - 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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10.1098/rspa.2023.0828

rspa.2023.0828.pdf
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.
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Cite this

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title = "Periodic finite-band solutions to the focusing nonlinear Schr{\"o}dinger 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 Schr{\"o}dinger (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.",

keywords = "periodic finite-band solutions, Riemann–Hilbert problem, Fokas method, nonlinear Schr{\"o}dinger equation",

author = "Dmitry Shepelsky and Iryna Karpenko and Stepan Bogdanov and Prilepsky, {Jaroslaw E.}",

note = "Copyright {\textcopyright} 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.",

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AU - Shepelsky, Dmitry

AU - Karpenko, Iryna

AU - Bogdanov, Stepan

AU - Prilepsky, Jaroslaw E.

N1 - 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.

PY - 2024/3/27

Y1 - 2024/3/27

N2 - 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.

AB - 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.

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

Abstract

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Keywords

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