Self-Similar Singular Solutions to the Nonlinear Schrödinger and the Complex Ginzburg-Landau Equations
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909435617607680 |
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| author | Dahne, Joel Figueras, Jordi-Lluís |
| author_facet | Dahne, Joel Figueras, Jordi-Lluís |
| contents | We prove the existence of radial self-similar singular solutions for the mass supercritical Nonlinear Schrödinger Equation far from the critical regime and, more generally, branches of such solutions for the Complex Ginzburg-Landau Equation. We are also able to control their monotone index (number of monotone intervals). In particular, we prove the existence of monotone radial self-similar singular solutions for the three dimensional cubic Nonlinear Schrödinger Equation. The paper combines sharp analytic bounds of the self-similar profile at infinity with computer assisted bounds around zero and their matching at an intermediate value. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_05480 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Self-Similar Singular Solutions to the Nonlinear Schrödinger and the Complex Ginzburg-Landau Equations Dahne, Joel Figueras, Jordi-Lluís Analysis of PDEs We prove the existence of radial self-similar singular solutions for the mass supercritical Nonlinear Schrödinger Equation far from the critical regime and, more generally, branches of such solutions for the Complex Ginzburg-Landau Equation. We are also able to control their monotone index (number of monotone intervals). In particular, we prove the existence of monotone radial self-similar singular solutions for the three dimensional cubic Nonlinear Schrödinger Equation. The paper combines sharp analytic bounds of the self-similar profile at infinity with computer assisted bounds around zero and their matching at an intermediate value. |
| title | Self-Similar Singular Solutions to the Nonlinear Schrödinger and the Complex Ginzburg-Landau Equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2410.05480 |