# Randomized exponential integrators for modulated nonlinear Schrödinger equations

Hofmanová, Martina; Knöller, Marvin; Schratz, Katharina

##### Abstract:
We consider the nonlinear Schrödinger equation with dispersion modulated by a (formal) derivative of a $\alpha$-Hölder continuous time-dependent function. Due to the highly oscillatory nature of the problem classical numerical methods face severe order reduction in non-smooth regimes $\alpha < 1$. In this work, we develop a new randomized exponential integrator based on a stratified Monte Carlo approximation which allows us to average the high oscillations in the problem and obtain improved error bounds of order $\alpha + 1/2$. In addition, the new approach allows us to treat a far more general class of modulations than the available literature. Numerical results underline our theoretical findings and show the favorable error behavior of our new scheme compared to classical methods.

 Zugehörige Institution(en) am KIT Institut für Angewandte und Numerische Mathematik (IANM)Sonderforschungsbereich 1173 (SFB 1173) Publikationstyp Forschungsbericht Jahr 2018 Sprache Englisch Identifikator ISSN: 2365-662X urn:nbn:de:swb:90-886009 KITopen-ID: 1000088600 Verlag KIT, Karlsruhe Umfang 17 S. Serie CRC 1173 ; 2018/49 Projektinformation SFB 1173/1 (DFG, DFG KOORD, SFB 1173/1 2015) Schlagworte modulated Schrödinger equation, order reduction, highly oscillatory problems, exponential-type integrator
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