Section: New Results

Stroboscopic averaging for the nonlinear Schrödinger equation

In [18] , we are concerned with an averaging procedure, -namely Stroboscopic averaging-, 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 high-order 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 Schrd̎inger equation on the d-dimensional torus $T^d$, or in $R^d$ 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.