Stroboscopic Averaging for the Nonlinear Schrdinger Equation
In this paper, 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 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 Schrdinger equation on the \(d\)-dimensional torus \(\mathbb {T}^d\), or in \(\mathbb {R}^d\) with a harmonic oscillator, for which we obtain a hierarchy of Hamiltonian averaged models. Our results are then illustrated numerically on several examples borrowed from the recent literature.
stroboscopic averaging nonlinear schrdinger highly oscillatory evolution equation invariants hamiltonian pdes sam 35a35 35j10 34k33