Stroboscopic Averaging for the Nonlinear Schrödinger Equation

International audienceIn 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

Time-integration methods for a dispersion-managed nonlinear Schrödinger equation

Modeling dispersion-managed optical fibers leads to a nonlinear Schrödinger equation where the linear part is multiplied by a rapidly changing piecewise constant coefficient function. Typically, the occurring oscillations of the solution and the discontinuous coefficients impose severe problems for traditional time-integrators. In this thesis, we present and analyze tailor-made numerical methods for this equation which attain a desired accuracy with significantly larger step-sizes than traditional methods. The construction of the methods is based on a favorable transformation of problem and the explicit computation of certain integrals over highly oscillatory phases. In the error analysis, we deviate from the classical concept “stability and consistency yield convergence”. Instead, we utilize recursion formulas for the global error to exploit cancellation effects of various oscillatory error terms allowing us to prove higher accuracy for special step-sizes