Numerical computation of solitons for optical systems

Laurent Di Menza

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • 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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Di Menza, Laurent. "Numerical computation of solitons for optical systems." ESAIM: Mathematical Modelling and Numerical Analysis 43.1 (2008): 173-208. <http://eudml.org/doc/194443>.

@article{DiMenza2008,
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 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. },
author = {Di Menza, Laurent},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Nonlinear optics; elliptic problems; stationary states; shooting method; continuation method.; nonlinear optics; continuation method; numerical examples; solitons; nonlinear Schrödinger equation},
language = {eng},
month = {12},
number = {1},
pages = {173-208},
publisher = {EDP Sciences},
title = {Numerical computation of solitons for optical systems},
url = {http://eudml.org/doc/194443},
volume = {43},
year = {2008},
}

TY - JOUR
AU - Di Menza, Laurent
TI - Numerical computation of solitons for optical systems
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/12//
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 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.
LA - eng
KW - Nonlinear optics; elliptic problems; stationary states; shooting method; continuation method.; nonlinear optics; continuation method; numerical examples; solitons; nonlinear Schrödinger equation
UR - http://eudml.org/doc/194443
ER -

References

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