Large Deviations for the Nonlinear Schr\"odinger Equation with Randomized Quasi-Periodic Initial Data in Higher Dimensions: Subcritical Case

Abstract

We study the cubic weakly nonlinear Schr\"odinger equation with randomized spatially quasi-periodic initial data in higher dimensions. Under a polynomial decay assumption in Fourier space, we establish a \emph{Large Deviations Principle} for rogue waves in the so-called \emph{subcritical} time regime. The proof proceeds in two main steps. We first characterize the distribution of the linear solution and establish the corresponding linear large deviations principle. The lower bound is obtained via pointwise estimates, while the upper bound follows from a combination of truncation and probabilistic arguments. {The method used in this step appears to be {\em new}; compare with \cite{GGKS23}.} We then perform a detailed combinatorial analysis of the Picard iteration, deriving an effective bound for the Duhamel term and thereby establishing the nonlinear large deviations principle.