An explicitly solvable NLS model with discontinuous standing waves
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866912636982001664 |
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| author | Adami, Riccardo Boni, Filippo Nakamura, Takaaki Ruighi, Alice |
| author_facet | Adami, Riccardo Boni, Filippo Nakamura, Takaaki Ruighi, Alice |
| contents | We study the NLS Equation on the line with a point interaction given by the superposition of an attractive delta potential with a dipole interaction, in the cases of $L^2$-subcritical and $L^2$-critical nonlinearity. For a subcritical nonlinearity we prove the existence and the uniqueness of Ground States at any mass. If the mass exceeds an explicit threshold, then there exists a positive excited state too. For the critical nonlinearity we prove that Ground States exist only in a specific interval of masses, while in a different interval excited states exist. We provide the value of the optimal constant in the Gagliardo-Nirenberg estimate and describe in the dipole case the branches of the stationary states as the strength of the interaction varies. Since all stationary states are explicitly computed, ours is a solvable model involving a non-standard interplay of a nonlinearity with a point interaction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_03374 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An explicitly solvable NLS model with discontinuous standing waves Adami, Riccardo Boni, Filippo Nakamura, Takaaki Ruighi, Alice Analysis of PDEs We study the NLS Equation on the line with a point interaction given by the superposition of an attractive delta potential with a dipole interaction, in the cases of $L^2$-subcritical and $L^2$-critical nonlinearity. For a subcritical nonlinearity we prove the existence and the uniqueness of Ground States at any mass. If the mass exceeds an explicit threshold, then there exists a positive excited state too. For the critical nonlinearity we prove that Ground States exist only in a specific interval of masses, while in a different interval excited states exist. We provide the value of the optimal constant in the Gagliardo-Nirenberg estimate and describe in the dipole case the branches of the stationary states as the strength of the interaction varies. Since all stationary states are explicitly computed, ours is a solvable model involving a non-standard interplay of a nonlinearity with a point interaction. |
| title | An explicitly solvable NLS model with discontinuous standing waves |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2502.03374 |