Numerical methods for a class of generalized nonlinear Schrödinger equations

Shalva Amiranashvili; Raimondas  Čiegis; Mindaugas Radziunas

doi:10.3934/krm.2015.8.215

Article Contents

2015, Volume 8, Issue 2: 215-234. Doi: 10.3934/krm.2015.8.215

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Numerical methods for a class of generalized nonlinear Schrödinger equations

1.
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, 10117 Berlin
2.
Vilnius Gediminas Technical University, Saulėtekio al. 11, LT-10223 Vilnius

Received: July 31, 2013

Revised: August 31, 2014

Published: May 2015

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Abstract

We present and analyze different splitting algorithms for numerical solution of the both classical and generalized nonlinear Schrödinger equations describing propagation of wave packets with special emphasis on applications to nonlinear fiber-optics. The considered generalizations take into account the higher-order corrections of the linear differential dispersion operator as well as the saturation of nonlinearity and the self-steepening of the field envelope function. For stabilization of the pseudo-spectral splitting schemes for generalized Schrödinger equations a regularization based on the approximation of the derivatives by the low number of Fourier modes is proposed. To illustrate the theoretically predicted performance of these schemes several numerical experiments have been done. In particular, we compute real-world examples of extreme pulses propagating in silica fibers.

Keywords:

Generalized nonlinear Schrödinger equations,
finite diiference method,
splitting method,
spectral method.

Mathematics Subject Classification: Primary: 65M60, 65M70; Secondary: 35G31.

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References

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