Periodic finite-band solutions to the focusing nonlinear Schrödinger equation by the Fokas method: inverse and direct problems
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866910390391144448 |
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| author | Shepelsky, Dmitry Karpenko, Iryna Bogdanov, Stepan Prilepsky, Jaroslaw E. |
| author_facet | Shepelsky, Dmitry Karpenko, Iryna Bogdanov, Stepan Prilepsky, Jaroslaw E. |
| contents | 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 [1,2]. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_16902 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Periodic finite-band solutions to the focusing nonlinear Schrödinger equation by the Fokas method: inverse and direct problems Shepelsky, Dmitry Karpenko, Iryna Bogdanov, Stepan Prilepsky, Jaroslaw E. Exactly Solvable and Integrable Systems Analysis of PDEs 35Q55, 35Q15, 37K15 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 [1,2]. 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. |
| title | Periodic finite-band solutions to the focusing nonlinear Schrödinger equation by the Fokas method: inverse and direct problems |
| topic | Exactly Solvable and Integrable Systems Analysis of PDEs 35Q55, 35Q15, 37K15 |
| url | https://arxiv.org/abs/2311.16902 |