On periodic solutions and attractors for the Maxwell--Bloch equations
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866912759940120576 |
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| author | Komech, Alexander |
| author_facet | Komech, Alexander |
| contents | We consider the Maxwell-Bloch system which is a finite-dimensional approximation of the coupled nonlinear Maxwell-Schrödinger equations. The approximation consists of one-mode Maxwell field coupled to two-level molecule. We construct time-periodic solutions to the factordynamics which is due to the symmetry gauge group. For the corresponding solutions to the Maxwell--Bloch system, the Maxwell field, current and the population inversion are time-periodic, while the wave function acquires a unit factor in the period. The proofs rely on high-amplitude asymptotics of the Maxwell field and a suitable extension of the Lefschetz theorem on fixed points and the Euler characteristic for noncompact manifolds. We also prove the existence of the global compact attractor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_08180 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On periodic solutions and attractors for the Maxwell--Bloch equations Komech, Alexander Mathematical Physics 34C25, 34C14, 57Q65, 32S50, 78A40, 78A60 We consider the Maxwell-Bloch system which is a finite-dimensional approximation of the coupled nonlinear Maxwell-Schrödinger equations. The approximation consists of one-mode Maxwell field coupled to two-level molecule. We construct time-periodic solutions to the factordynamics which is due to the symmetry gauge group. For the corresponding solutions to the Maxwell--Bloch system, the Maxwell field, current and the population inversion are time-periodic, while the wave function acquires a unit factor in the period. The proofs rely on high-amplitude asymptotics of the Maxwell field and a suitable extension of the Lefschetz theorem on fixed points and the Euler characteristic for noncompact manifolds. We also prove the existence of the global compact attractor. |
| title | On periodic solutions and attractors for the Maxwell--Bloch equations |
| topic | Mathematical Physics 34C25, 34C14, 57Q65, 32S50, 78A40, 78A60 |
| url | https://arxiv.org/abs/2312.08180 |