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Large sets of Anderson localized states exist for the quasi-periodic nonlinear Schrödinger equation on $\mathbb Z^d$, demonstrating that Anderson localization occurs in both nonlinear and deterministic settings. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

June 7, 2026Open Access

Anderson localized states for the quasi-periodic nonlinear Schrödinger equation on Z d Zᵈdouble struck upper Z Superscript d

Key Points

The research aims to establish large sets of Anderson localized states for the quasi-periodic nonlinear Schrödinger equation.
Extending Anderson localization from linear to nonlinear contexts.
Developing a new Diophantine estimate for quasi-periodic functions in arbitrary-dimensional phase space.
Applying Bourgain’s geometric lemma for analysis.
Demonstrated Anderson localization in a deterministic setting for nonlinear equations.
Validated the existence of large sets of localized states in higher-dimensional systems.

Abstract

Abstract We establish large sets of Anderson localized states for the quasi-periodic nonlinear Schrödinger equation on Z d Zᵈ double struck upper Z Superscript d, thus extending Anderson localization from the linear (cf. Bourgain Geom. Funct. Anal. , 17 (3): 682–706, 2007) to a nonlinear setting, and from 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 arbitrary-dimensional phase space, and the application of Bourgain’s geometric lemma in Geom. Funct. Anal. , 17 (3): 682–706, 2007.

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