Stroboscopic Averaging for the Nonlinear Schrödinger Equation

In this paper, we are concerned with an averaging procedure, – namely Stroboscopic averaging [SVM07, CMSS10] –, for highly-oscillatory evolution equations posed in a (possibly infinite dimensional) Banach space, typically partial differential equations (PDEs) in a high-frequency regime where only one frequency is present. We construct a highorder averaged system whose solution remains exponentially close to the exact one over long time intervals, possesses the same geometric properties (structure, invariants, . . . ) as compared to the original system, and is non-oscillatory. We then apply our results to the nonlinear Schrödinger equation on the d-dimensional torus T, or in R with a harmonic oscillator, for which we obtain a hierarchy of Hamiltonian averaged models. Our results are illustrated numerically on several examples borrowed from the recent literature.