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

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].