Initial-Boundary Value Problem for the Maxwell-Bloch Equations with an Arbitrary Inhomogeneous Broadening and Periodic Boundary Function
The initial-boundary value problem (IBVP) for the Maxwell-Bloch equations with an arbitrary inhomogeneous broadening and periodic boundary conditions is studied. This IBVP describes the propagation of an electromagnetic wave generated by periodic pumping in a resonant medium with distributed two-lev...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2023 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2023
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/212035 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Initial-Boundary Value Problem for the Maxwell-Bloch Equations with an Arbitrary Inhomogeneous Broadening and Periodic Boundary Function. Maria Filipkovska. SIGMA 19 (2023), 096, 39 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862543423904415744 |
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| author | Filipkovska, Maria |
| author_facet | Filipkovska, Maria |
| citation_txt | Initial-Boundary Value Problem for the Maxwell-Bloch Equations with an Arbitrary Inhomogeneous Broadening and Periodic Boundary Function. Maria Filipkovska. SIGMA 19 (2023), 096, 39 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The initial-boundary value problem (IBVP) for the Maxwell-Bloch equations with an arbitrary inhomogeneous broadening and periodic boundary conditions is studied. This IBVP describes the propagation of an electromagnetic wave generated by periodic pumping in a resonant medium with distributed two-level atoms. We extended the inverse scattering transform method in the form of the matrix Riemann-Hilbert problem for solving the considered IBVP. Using the system of Ablowitz-Kaup-Newell-Segur equations equivalent to the system of the Maxwell-Bloch (MB) equations, we construct the associated matrix Riemann-Hilbert (RH) problem. Theorems on the existence, uniqueness, and smoothness properties of a solution of the constructed RH problem are proved, and it is shown that the solution of the associated RH problem uniquely defines a solution of the considered IBVP. It is proven that the RH problem provides the causality principle. The representation of a solution of the MB equations in terms of a solution of the associated RH problem is given. The significance of this method also lies in the fact that, having studied the asymptotic behavior of the constructed RH problem and equivalent ones, we can obtain formulas for the asymptotics of a solution of the corresponding IBVP for the MB equations.
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| first_indexed | 2026-03-12T22:05:19Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212035 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-12T22:05:19Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Filipkovska, Maria 2026-01-23T10:09:31Z 2023 Initial-Boundary Value Problem for the Maxwell-Bloch Equations with an Arbitrary Inhomogeneous Broadening and Periodic Boundary Function. Maria Filipkovska. SIGMA 19 (2023), 096, 39 pages 1815-0659 2020 Mathematics Subject Classification: 35F31; 35Q15; 37K15; 34L25; 35Q60 arXiv:2212.04524 https://nasplib.isofts.kiev.ua/handle/123456789/212035 https://doi.org/10.3842/SIGMA.2023.096 The initial-boundary value problem (IBVP) for the Maxwell-Bloch equations with an arbitrary inhomogeneous broadening and periodic boundary conditions is studied. This IBVP describes the propagation of an electromagnetic wave generated by periodic pumping in a resonant medium with distributed two-level atoms. We extended the inverse scattering transform method in the form of the matrix Riemann-Hilbert problem for solving the considered IBVP. Using the system of Ablowitz-Kaup-Newell-Segur equations equivalent to the system of the Maxwell-Bloch (MB) equations, we construct the associated matrix Riemann-Hilbert (RH) problem. Theorems on the existence, uniqueness, and smoothness properties of a solution of the constructed RH problem are proved, and it is shown that the solution of the associated RH problem uniquely defines a solution of the considered IBVP. It is proven that the RH problem provides the causality principle. The representation of a solution of the MB equations in terms of a solution of the associated RH problem is given. The significance of this method also lies in the fact that, having studied the asymptotic behavior of the constructed RH problem and equivalent ones, we can obtain formulas for the asymptotics of a solution of the corresponding IBVP for the MB equations. The author would like to thank Vladimir Kotlyarov (B. Verkin ILTPE of NAS of Ukraine) for useful discussions and suggestions. Also, the author would like to thank the anonymous referees for their careful reading of the manuscript and their comments. This work was partially supported by the National Academy of Sciences of Ukraine (project 0121U111968 “Nonstandard nonlocal and peakon integrable equations: asymptotics and inverse scattering transform”) and the Alexander von Humboldt Foundation (the host institution: Friedrich-Alexander University of Erlangen-Nuremberg, Chair for Dynamics, Control, Machine Learning and Numerics). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Initial-Boundary Value Problem for the Maxwell-Bloch Equations with an Arbitrary Inhomogeneous Broadening and Periodic Boundary Function Article published earlier |
| spellingShingle | Initial-Boundary Value Problem for the Maxwell-Bloch Equations with an Arbitrary Inhomogeneous Broadening and Periodic Boundary Function Filipkovska, Maria |
| title | Initial-Boundary Value Problem for the Maxwell-Bloch Equations with an Arbitrary Inhomogeneous Broadening and Periodic Boundary Function |
| title_full | Initial-Boundary Value Problem for the Maxwell-Bloch Equations with an Arbitrary Inhomogeneous Broadening and Periodic Boundary Function |
| title_fullStr | Initial-Boundary Value Problem for the Maxwell-Bloch Equations with an Arbitrary Inhomogeneous Broadening and Periodic Boundary Function |
| title_full_unstemmed | Initial-Boundary Value Problem for the Maxwell-Bloch Equations with an Arbitrary Inhomogeneous Broadening and Periodic Boundary Function |
| title_short | Initial-Boundary Value Problem for the Maxwell-Bloch Equations with an Arbitrary Inhomogeneous Broadening and Periodic Boundary Function |
| title_sort | initial-boundary value problem for the maxwell-bloch equations with an arbitrary inhomogeneous broadening and periodic boundary function |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212035 |
| work_keys_str_mv | AT filipkovskamaria initialboundaryvalueproblemforthemaxwellblochequationswithanarbitraryinhomogeneousbroadeningandperiodicboundaryfunction |