Numerical computation of solitons for optical systems

Laurent Di Menza

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2009)

  • Volume: 43, Issue: 1, page 173-208
  • ISSN: 0764-583X

Abstract

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In this paper, we present numerical methods for the determination of solitons, that consist in spatially localized stationary states of nonlinear scalar equations or coupled systems arising in nonlinear optics. We first use the well-known shooting method in order to find excited states (characterized by the number k of nodes) for the classical nonlinear Schrödinger equation. Asymptotics can then be derived in the limits of either large k are large nonlinear exponents σ . In a second part, we compute solitons for a nonlinear system governing the propagation of two coupled waves in a quadratic media in any spatial dimension, starting from one-dimensional states obtained with a shooting method and considering the dimension as a continuation parameter. Finally, we investigate the case of three wave mixing, for which the shooting method is not relevant.

How to cite

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Menza, Laurent Di. "Numerical computation of solitons for optical systems." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 43.1 (2009): 173-208. <http://eudml.org/doc/245244>.

@article{Menza2009,
abstract = {In this paper, we present numerical methods for the determination of solitons, that consist in spatially localized stationary states of nonlinear scalar equations or coupled systems arising in nonlinear optics. We first use the well-known shooting method in order to find excited states (characterized by the number $k$ of nodes) for the classical nonlinear Schrödinger equation. Asymptotics can then be derived in the limits of either large $k$ are large nonlinear exponents $\sigma $. In a second part, we compute solitons for a nonlinear system governing the propagation of two coupled waves in a quadratic media in any spatial dimension, starting from one-dimensional states obtained with a shooting method and considering the dimension as a continuation parameter. Finally, we investigate the case of three wave mixing, for which the shooting method is not relevant.},
author = {Menza, Laurent Di},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {nonlinear optics; elliptic problems; stationary states; shooting method; continuation method; numerical examples; solitons; nonlinear Schrödinger equation},
language = {eng},
number = {1},
pages = {173-208},
publisher = {EDP-Sciences},
title = {Numerical computation of solitons for optical systems},
url = {http://eudml.org/doc/245244},
volume = {43},
year = {2009},
}

TY - JOUR
AU - Menza, Laurent Di
TI - Numerical computation of solitons for optical systems
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2009
PB - EDP-Sciences
VL - 43
IS - 1
SP - 173
EP - 208
AB - In this paper, we present numerical methods for the determination of solitons, that consist in spatially localized stationary states of nonlinear scalar equations or coupled systems arising in nonlinear optics. We first use the well-known shooting method in order to find excited states (characterized by the number $k$ of nodes) for the classical nonlinear Schrödinger equation. Asymptotics can then be derived in the limits of either large $k$ are large nonlinear exponents $\sigma $. In a second part, we compute solitons for a nonlinear system governing the propagation of two coupled waves in a quadratic media in any spatial dimension, starting from one-dimensional states obtained with a shooting method and considering the dimension as a continuation parameter. Finally, we investigate the case of three wave mixing, for which the shooting method is not relevant.
LA - eng
KW - nonlinear optics; elliptic problems; stationary states; shooting method; continuation method; numerical examples; solitons; nonlinear Schrödinger equation
UR - http://eudml.org/doc/245244
ER -

References

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