Well-posedness of the periodic nonlinear Schrödinger equation with concentrated nonlinearity
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918171948089344 |
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| author | Lee, Jinyeop Rout, Andrew |
| author_facet | Lee, Jinyeop Rout, Andrew |
| contents | We study the solution theory of the nonlinear Schrödinger equation with a concentrated nonlinearity on the torus. In particular, we establish existence and uniqueness of global energy-conserving solutions for initial data in $H^1$. Our approach is based on two approximation schemes, namely the concentrated limit of a smoothed nonlinear Schrödinger equation and the inviscid limit of a concentrated complex Ginzburg--Landau equation. We also prove the existence and uniquness of solutions below the energy space. To our knowledge, this is the first rigorous solution theory for a periodic nonlinear Schrödinger equation with a concentrated nonlinearity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_00594 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Well-posedness of the periodic nonlinear Schrödinger equation with concentrated nonlinearity Lee, Jinyeop Rout, Andrew Analysis of PDEs Mathematical Physics 35Q55, 35B25, 35A01, 35A02, 45D05 We study the solution theory of the nonlinear Schrödinger equation with a concentrated nonlinearity on the torus. In particular, we establish existence and uniqueness of global energy-conserving solutions for initial data in $H^1$. Our approach is based on two approximation schemes, namely the concentrated limit of a smoothed nonlinear Schrödinger equation and the inviscid limit of a concentrated complex Ginzburg--Landau equation. We also prove the existence and uniquness of solutions below the energy space. To our knowledge, this is the first rigorous solution theory for a periodic nonlinear Schrödinger equation with a concentrated nonlinearity. |
| title | Well-posedness of the periodic nonlinear Schrödinger equation with concentrated nonlinearity |
| topic | Analysis of PDEs Mathematical Physics 35Q55, 35B25, 35A01, 35A02, 45D05 |
| url | https://arxiv.org/abs/2508.00594 |