Numerische Mathematik

, Volume 138, Issue 4, pp 975–1009 | Cite as

Adiabatic midpoint rule for the dispersion-managed nonlinear Schrödinger equation

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Abstract

The dispersion-managed nonlinear Schrödinger equation contains a small parameter \(\varepsilon \), a rapidly changing piecewise constant coefficient function, and a cubic nonlinearity. Typical solutions are highly oscillatory and have a discontinuous time-derivative, and hence solving this equation numerically is a challenging task. We present and analyze a tailor-made time integrator which attains the desired accuracy with a significantly larger step-size than traditional methods. The construction of this method is based on a favorable transformation to an equivalent problem and the explicit computation of certain integrals over highly oscillatory phases. The error analysis requires the thorough investigation of various cancellation effects which result in improved accuracy for special step-sizes.

Mathematics Subject Classification

65M12 65M15 65M70 65Z05 35B40 35Q55 

Notes

Acknowledgements

We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173. Moreover, we thank the anonymous referee for many helpful comments.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Institute for Applied and Numerical MathematicsKarlsruhe Institute of TechnologyKarlsruheGermany

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