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Anderson localized states for the quasi-periodic nonlinear Schrödinger equation on $\mathbb Z^d$

Published 27 May 2024 in math-ph, math.AP, math.DS, and math.MP | (2405.17513v1)

Abstract: We establish large sets of Anderson localized states for the quasi-periodic nonlinear Schr\"odinger equation on $\mathbb Zd$, thus extending Anderson localization from the linear (cf. Bourgain [GAFA 17(3), 682--706, 2007]) to a nonlinear setting, and the random (cf. Bourgain-Wang [JEMS 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 [GAFA 17(3), 682--706, 2007].

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