Justification of the Asymptotic Coupled Mode Approximation of Out-of-Plane Gap Solitons in Maxwell Equations
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2020
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| _version_ | 1866912496836673536 |
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| author | Dohnal, Tomáš Romani, Giulio |
| author_facet | Dohnal, Tomáš Romani, Giulio |
| contents | In periodic media gap solitons with frequencies inside a spectral gap but close to a spectral band can be formally approximated by a slowly varying envelope ansatz. The ansatz is based on the linear Bloch waves at the edge of the band and on effective coupled mode equations (CMEs) for the envelopes. We provide a rigorous justification of such CME asymptotics in two-dimensional photonic crystals described by the Kerr nonlinear Maxwell system. We use a Lyapunov-Schmidt reduction procedure and a nested fixed point argument in the Bloch variables. The theorem provides an error estimate in $H^2(\mathbb R^2)$ between the exact solution and the envelope approximation. The results justify the formal and numerical CME-approximation in [Dohnal and Dörfler, Multiscale Model. Simul., p. 162-191, 11 (2013)]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_03473 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Justification of the Asymptotic Coupled Mode Approximation of Out-of-Plane Gap Solitons in Maxwell Equations Dohnal, Tomáš Romani, Giulio Analysis of PDEs 35Q61 35C08 35B32 35B06 35J61 In periodic media gap solitons with frequencies inside a spectral gap but close to a spectral band can be formally approximated by a slowly varying envelope ansatz. The ansatz is based on the linear Bloch waves at the edge of the band and on effective coupled mode equations (CMEs) for the envelopes. We provide a rigorous justification of such CME asymptotics in two-dimensional photonic crystals described by the Kerr nonlinear Maxwell system. We use a Lyapunov-Schmidt reduction procedure and a nested fixed point argument in the Bloch variables. The theorem provides an error estimate in $H^2(\mathbb R^2)$ between the exact solution and the envelope approximation. The results justify the formal and numerical CME-approximation in [Dohnal and Dörfler, Multiscale Model. Simul., p. 162-191, 11 (2013)]. |
| title | Justification of the Asymptotic Coupled Mode Approximation of Out-of-Plane Gap Solitons in Maxwell Equations |
| topic | Analysis of PDEs 35Q61 35C08 35B32 35B06 35J61 |
| url | https://arxiv.org/abs/2010.03473 |