Anderson localized states for the quasi-periodic nonlinear Schrödinger equation on $\mathbb Z^d$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866908971500044288 |
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| author | Shi, Yunfeng Wang, W. -M. |
| author_facet | Shi, Yunfeng Wang, W. -M. |
| contents | We establish large sets of Anderson localized states for the quasi-periodic nonlinear Schrödinger equation on $\mathbb Z^d$, thus extending Anderson localization from the linear (cf. Bourgain [Geom. Funct. Anal., 17(3):682--706, 2007]) to a nonlinear setting, and the random (cf. Bourgain-Wang [J. Eur. Math. Soc., 10(1):1--45, 2008]) to a deterministic setting. Among the main ingredients are a new Diophantine estimate of quasi-periodic functions in arbitrarily dimensional phase space, and the application of Bourgain's geometric lemma in [Geom. Funct. Anal., 17(3):682--706, 2007]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_17513 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Anderson localized states for the quasi-periodic nonlinear Schrödinger equation on $\mathbb Z^d$ Shi, Yunfeng Wang, W. -M. Mathematical Physics Analysis of PDEs Dynamical Systems We establish large sets of Anderson localized states for the quasi-periodic nonlinear Schrödinger equation on $\mathbb Z^d$, thus extending Anderson localization from the linear (cf. Bourgain [Geom. Funct. Anal., 17(3):682--706, 2007]) to a nonlinear setting, and the random (cf. Bourgain-Wang [J. Eur. Math. Soc., 10(1):1--45, 2008]) to a deterministic setting. Among the main ingredients are a new Diophantine estimate of quasi-periodic functions in arbitrarily dimensional phase space, and the application of Bourgain's geometric lemma in [Geom. Funct. Anal., 17(3):682--706, 2007]. |
| title | Anderson localized states for the quasi-periodic nonlinear Schrödinger equation on $\mathbb Z^d$ |
| topic | Mathematical Physics Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2405.17513 |