A note on the existence of self-similar profiles of the hydrodynamic formulation of the focusing nonlinear Schrödinger equation
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915207839744000 |
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| author | Cao-Labora, Gonzalo Gómez-Serrano, Javier Shi, Jia Staffilani, Gigliola |
| author_facet | Cao-Labora, Gonzalo Gómez-Serrano, Javier Shi, Jia Staffilani, Gigliola |
| contents | After performing the Madelung transformation, the nonlinear Schrödinger equation is transformed into a hydrodynamic equation akin to the compressible Euler equations with a certain dissipation. In this short note, we construct self-similar solutions of such system in the focusing case for any mass supercritical exponent. To the best of our knowledge these solutions are new, and may formally arise as potential blow-up profiles of the focusing NLS equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_16813 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on the existence of self-similar profiles of the hydrodynamic formulation of the focusing nonlinear Schrödinger equation Cao-Labora, Gonzalo Gómez-Serrano, Javier Shi, Jia Staffilani, Gigliola Analysis of PDEs After performing the Madelung transformation, the nonlinear Schrödinger equation is transformed into a hydrodynamic equation akin to the compressible Euler equations with a certain dissipation. In this short note, we construct self-similar solutions of such system in the focusing case for any mass supercritical exponent. To the best of our knowledge these solutions are new, and may formally arise as potential blow-up profiles of the focusing NLS equation. |
| title | A note on the existence of self-similar profiles of the hydrodynamic formulation of the focusing nonlinear Schrödinger equation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.16813 |