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Numerical computation of solitons for optical systems

Published online by Cambridge University Press:  05 December 2008

Laurent Di Menza
Laurent Di Menza*
Affiliation:
Université de Reims, Laboratoire de Mathématiques, Moulin de la Housse, BP 1039, 51687 Reims Cedex 2, France. [email protected]
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Abstract

In this paper, we present numerical methodsfor the determination of solitons, that consist in spatially localizedstationary states of nonlinear scalar equations or coupled systemsarising in nonlinear optics.We first use the well-known shooting method in order to findexcited states (characterized by the number k of nodes) for theclassical nonlinear Schrödinger equation. Asymptotics can thenbe derived in the limits of either large k are large nonlinear exponents σ.In a second part, we compute solitons for a nonlinearsystem governing the propagation of two coupled waves in a quadratic media in anyspatial 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 wavemixing, for which the shooting method is not relevant.

Keywords

Nonlinear opticselliptic problemsstationary statesshooting methodcontinuation method.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 43 , Issue 1 , January 2009 , pp. 173 - 208
Copyright
© EDP Sciences, SMAI, 2008

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