Numerical computation of solitons for optical systems

Menza, Laurent Di

doi:10.1051/m2an:2008044

Menza, Laurent Di

ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 1, pp. 173-208.

Résumé

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.

MR Zbl

DOI : 10.1051/m2an:2008044

Classification : 35J25, 35J60, 65N06, 65N99, 78M20
Mots clés : nonlinear optics, elliptic problems, stationary states, shooting method, continuation method

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     title = {Numerical computation of solitons for optical systems},
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     publisher = {EDP-Sciences},
     volume = {43},
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Menza, Laurent Di. Numerical computation of solitons for optical systems. ESAIM: Modélisation mathématique et analyse numérique, Tome 43 (2009) no. 1, pp. 173-208. doi : 10.1051/m2an:2008044. http://archive.numdam.org/articles/10.1051/m2an:2008044/

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