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Journal Articles Foundations of Computational Mathematics Year : 2015

Stroboscopic Averaging for the Nonlinear Schrödinger Equation

Abstract

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 $T^d$, 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.
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Dates and versions

hal-00732850 , version 1 (17-09-2012)

Identifiers

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François Castella, Philippe Chartier, Florian Méhats, Ander Murua. Stroboscopic Averaging for the Nonlinear Schrödinger Equation. Foundations of Computational Mathematics, 2015, 15 (2), pp.519-559. ⟨10.1007/s10208-014-9235-7⟩. ⟨hal-00732850⟩
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