Well-posedness and exponential stability of dispersive nonlinear Maxwell equations with PML: An evolutionary approach
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912147744751616 |
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| author | Margenberg, Nils Bause, Markus |
| author_facet | Margenberg, Nils Bause, Markus |
| contents | This paper presents a mathematical foundation for physical models in nonlinear optics through the lens of evolutionary equations. It focuses on two key concepts: well-posedness and exponential stability of Maxwell equations, with models that include materials with complex dielectric properties, dispersion, and discontinuities. We use a Hilbert space framework to address these complex physical models in nonlinear optics. While our focus is on the first-order formulation in space and time, higher solution regularity recovers and equates to the second-order formulation. We incorporate perfectly matched layers (PMLs), which model absorbing boundary conditions, to facilitate the development of numerical methods. We demonstrate that the combined system remains well-posed and exponentially stable. Our approach applies to a broad class of partial differential equations (PDEs) and accommodates materials with nonlocal behavior in space and time. The contribution of this work is a unified framework for analyzing wave interactions in advanced optical materials. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_05468 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Well-posedness and exponential stability of dispersive nonlinear Maxwell equations with PML: An evolutionary approach Margenberg, Nils Bause, Markus Analysis of PDEs Mathematical Physics 35Q61, 35B35, 35A01, 35A02, 35F20 This paper presents a mathematical foundation for physical models in nonlinear optics through the lens of evolutionary equations. It focuses on two key concepts: well-posedness and exponential stability of Maxwell equations, with models that include materials with complex dielectric properties, dispersion, and discontinuities. We use a Hilbert space framework to address these complex physical models in nonlinear optics. While our focus is on the first-order formulation in space and time, higher solution regularity recovers and equates to the second-order formulation. We incorporate perfectly matched layers (PMLs), which model absorbing boundary conditions, to facilitate the development of numerical methods. We demonstrate that the combined system remains well-posed and exponentially stable. Our approach applies to a broad class of partial differential equations (PDEs) and accommodates materials with nonlocal behavior in space and time. The contribution of this work is a unified framework for analyzing wave interactions in advanced optical materials. |
| title | Well-posedness and exponential stability of dispersive nonlinear Maxwell equations with PML: An evolutionary approach |
| topic | Analysis of PDEs Mathematical Physics 35Q61, 35B35, 35A01, 35A02, 35F20 |
| url | https://arxiv.org/abs/2412.05468 |