Numerical computation of solitons for optical systems

Université de Reims, Laboratoire de Mathématiques, Moulin de la Housse, BP 1039, 51687 Reims Cedex 2, France. laurent.di-menza@univ-reims.fr

Received: 14 July 2006
Revised: 7 April 2008

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.