A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions
The Baker-Akhiezer (BA) function theory was successfully developed in the mid-1970s. This theory brought very interesting and important results in the spectral theory of almost periodic operators and the theory of completely integrable nonlinear equations, such as the Korteweg-de Vries equation, the...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2018 |
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| Sprache: | Englisch |
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Інститут математики НАН України
2018
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/209768 |
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| Zitieren: | A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions / V.P. Kotlyarov // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 64 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862719064805212160 |
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| author | Kotlyarov, V.P. |
| author_facet | Kotlyarov, V.P. |
| citation_txt | A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions / V.P. Kotlyarov // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 64 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The Baker-Akhiezer (BA) function theory was successfully developed in the mid-1970s. This theory brought very interesting and important results in the spectral theory of almost periodic operators and the theory of completely integrable nonlinear equations, such as the Korteweg-de Vries equation, the nonlinear Schrödinger equation, the sine-Gordon equation, Kadomtsev-Petviashvili equation. Subsequently, the theory was reproduced for the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchies. However, extensions of the Baker-Akhiezer function for the Maxwell-Bloch (MB) system or for the Karpman-Kaup equations, which contain prescribed weight functions characterizing inhomogeneous broadening of the main frequency, are unknown. The main goal of the paper is to give a such of extension associated with the Maxwell-Bloch equations. Using different Riemann-Hilbert problems posed on the complex plane with a finite number of cuts, we propose such a matrix function that has a unit determinant and takes an explicit form through Cauchy integrals, hyperelliptic integrals, and theta functions. The matrix BA function solves the AKNS equations (the Lax pair for the MB system) and generates a quasi-periodic finite-gap solution to the Maxwell-Bloch equations. The suggested function will be useful in the study of the long-time asymptotic behavior of solutions of different initial-boundary value problems for the MB equations using the Deift-Zhou method of steepest descent and for an investigation of rogue waves of the Maxwell-Bloch equations.
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| first_indexed | 2025-12-07T18:18:45Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-209768 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T18:18:45Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
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| spelling | Kotlyarov, V.P. 2025-11-26T11:26:16Z 2018 A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions / V.P. Kotlyarov // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 64 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 34L25; 34M50; 35F31; 35Q15; 35Q51 arXiv: 1802.01622 https://nasplib.isofts.kiev.ua/handle/123456789/209768 https://doi.org/10.3842/SIGMA.2018.082 The Baker-Akhiezer (BA) function theory was successfully developed in the mid-1970s. This theory brought very interesting and important results in the spectral theory of almost periodic operators and the theory of completely integrable nonlinear equations, such as the Korteweg-de Vries equation, the nonlinear Schrödinger equation, the sine-Gordon equation, Kadomtsev-Petviashvili equation. Subsequently, the theory was reproduced for the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchies. However, extensions of the Baker-Akhiezer function for the Maxwell-Bloch (MB) system or for the Karpman-Kaup equations, which contain prescribed weight functions characterizing inhomogeneous broadening of the main frequency, are unknown. The main goal of the paper is to give a such of extension associated with the Maxwell-Bloch equations. Using different Riemann-Hilbert problems posed on the complex plane with a finite number of cuts, we propose such a matrix function that has a unit determinant and takes an explicit form through Cauchy integrals, hyperelliptic integrals, and theta functions. The matrix BA function solves the AKNS equations (the Lax pair for the MB system) and generates a quasi-periodic finite-gap solution to the Maxwell-Bloch equations. The suggested function will be useful in the study of the long-time asymptotic behavior of solutions of different initial-boundary value problems for the MB equations using the Deift-Zhou method of steepest descent and for an investigation of rogue waves of the Maxwell-Bloch equations. The author thanks the referees for careful reading of the manuscript and valuable recommendations. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions Article published earlier |
| spellingShingle | A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions Kotlyarov, V.P. |
| title | A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions |
| title_full | A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions |
| title_fullStr | A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions |
| title_full_unstemmed | A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions |
| title_short | A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions |
| title_sort | matrix baker-akhiezer function associated with the maxwell-bloch equations and their finite-gap solutions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209768 |
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