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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.01132 |
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| _version_ | 1866913971679789056 |
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| author | Fillman, Jake Li, Long Lukić, Milivoje Zhou, Qi |
| author_facet | Fillman, Jake Li, Long Lukić, Milivoje Zhou, Qi |
| contents | This paper addresses the Cauchy problem for the cubic defocusing nonlinear Schrödinger equation (NLS) with almost periodic initial data. We prove that for small analytic quasiperiodic initial data satisfying Diophantine frequency conditions, the Cauchy problem admits a solution that is almost periodic in both space and time, and that this solution is unique among solutions locally bounded in a suitable sense. The analysis combines direct and inverse spectral theory.
In the inverse spectral theory part, we prove existence, almost periodicity, and uniqueness for solutions with initial data whose associated Dirac operator has purely a.c.\ spectrum that is not too thin. This resolves novel challenges presented by the NLS hierarchy, such as an additional degree of freedom and an additional commuting flow.
In the direct spectral theory part, for Dirac operators with small analytic quasiperiodic potentials with Diophantine frequency conditions, we prove pure a.c.\ spectrum, exponentially decaying spectral gaps, and spectral thickness conditions (homogeneity and Craig-type conditions). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_01132 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Almost Periodic Solutions of The Cubic Defocusing Nonlinear Schrödinger Equation Fillman, Jake Li, Long Lukić, Milivoje Zhou, Qi Analysis of PDEs Mathematical Physics Dynamical Systems Spectral Theory 35Q55, 35R30, 47A10, 37K10, 47E05 This paper addresses the Cauchy problem for the cubic defocusing nonlinear Schrödinger equation (NLS) with almost periodic initial data. We prove that for small analytic quasiperiodic initial data satisfying Diophantine frequency conditions, the Cauchy problem admits a solution that is almost periodic in both space and time, and that this solution is unique among solutions locally bounded in a suitable sense. The analysis combines direct and inverse spectral theory. In the inverse spectral theory part, we prove existence, almost periodicity, and uniqueness for solutions with initial data whose associated Dirac operator has purely a.c.\ spectrum that is not too thin. This resolves novel challenges presented by the NLS hierarchy, such as an additional degree of freedom and an additional commuting flow. In the direct spectral theory part, for Dirac operators with small analytic quasiperiodic potentials with Diophantine frequency conditions, we prove pure a.c.\ spectrum, exponentially decaying spectral gaps, and spectral thickness conditions (homogeneity and Craig-type conditions). |
| title | Almost Periodic Solutions of The Cubic Defocusing Nonlinear Schrödinger Equation |
| topic | Analysis of PDEs Mathematical Physics Dynamical Systems Spectral Theory 35Q55, 35R30, 47A10, 37K10, 47E05 |
| url | https://arxiv.org/abs/2508.01132 |