Statistical solutions to the Schrödinger map equation in 1D, via the randomly forced Landau-Lifschitz-Gilbert equation
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915125699543040 |
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| author | Gussetti, Emanuela Hofmanová, Martina |
| author_facet | Gussetti, Emanuela Hofmanová, Martina |
| contents | We prove the existence of statistically stationary solutions to the Schrödinger map equation on a one-dimensional domain, with null Neumann boundary conditions. We deal directly with the equation in its real-valued formulation, without using any transform. To approximate the Schrödinger map equation, we employ the stochastic Landau-Lifschitz-Gilbert equation. By a limiting procedure à la Kuksin, we establish existence of a random initial datum, whose distribution is preserved under the dynamics of the deterministic equation. Among other properties, the corresponding statistically stationary solution is proved to exhibit non-trivial dynamics in space and time and to be genuinely random. With an analogous argument, we prove the existence of stationary solutions to a stochastic Schrödinger map equation. We discuss the relationship between the statistically stationary solutions to the Schrödinger map equation, the binormal curvature flow and the cubic non-linear Schrödinger equation. Additionally, we prove the existence of statistically stationary solutions to the binormal curvature flow. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_16499 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Statistical solutions to the Schrödinger map equation in 1D, via the randomly forced Landau-Lifschitz-Gilbert equation Gussetti, Emanuela Hofmanová, Martina Analysis of PDEs Mathematical Physics Probability 60G10, 60H15, 60L90, 60H30 We prove the existence of statistically stationary solutions to the Schrödinger map equation on a one-dimensional domain, with null Neumann boundary conditions. We deal directly with the equation in its real-valued formulation, without using any transform. To approximate the Schrödinger map equation, we employ the stochastic Landau-Lifschitz-Gilbert equation. By a limiting procedure à la Kuksin, we establish existence of a random initial datum, whose distribution is preserved under the dynamics of the deterministic equation. Among other properties, the corresponding statistically stationary solution is proved to exhibit non-trivial dynamics in space and time and to be genuinely random. With an analogous argument, we prove the existence of stationary solutions to a stochastic Schrödinger map equation. We discuss the relationship between the statistically stationary solutions to the Schrödinger map equation, the binormal curvature flow and the cubic non-linear Schrödinger equation. Additionally, we prove the existence of statistically stationary solutions to the binormal curvature flow. |
| title | Statistical solutions to the Schrödinger map equation in 1D, via the randomly forced Landau-Lifschitz-Gilbert equation |
| topic | Analysis of PDEs Mathematical Physics Probability 60G10, 60H15, 60L90, 60H30 |
| url | https://arxiv.org/abs/2501.16499 |