Hostname: page-component-848d4c4894-xm8r8 Total loading time: 0 Render date: 2024-07-08T03:21:12.808Z Has data issue: false hasContentIssue false
  • English
  • Français

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. laurent.di-menza@univ-reims.fr
Get access

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.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Balabane, M., Dolbeault, J. and Ounaies, H., Nodal solutions for a sublinear elliptic equation. Nonlinear Anal. 52 (2003) 219–237. CrossRef
Buryak, A.V., Steblina, V.V. and Kivshar, Y., Self-trapping of light beams and parametric solitons in diffractive quadratic media. Phys. Rev. A 52 (1995) 1670–1674. CrossRef
Buryak, A.V., Di Trapani, P., Skryabin, D.V. and Trillo, S., Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications. Phys. Rep. 370 (2002) 62–235.
Di Menza, L., Transparent and absorbing conditions for the Schrödinger equation in a bounded domain. Numer. Funct. Anal. Optim. 18 (1997) 759–775. CrossRef
Fibich, G., Gavish, N. and Wang, X.-P., Singular ring solutions of critical and supercritical nonlinear Schrödinger equations. Physica D 18 (2007) 55–86. CrossRef
Firth, W.J. and Skryabin, D.V., Optical solitons carrying orbital angular momentum. Phys. Rev. Lett. 79 (1997) 2450–2453. CrossRef
He, H., Werner, M.J. and Drummond, P.D., Simultaneous solitary-wave solutions in a nonlinear parametric waveguide. Phys. Rev. E 54 (1996) 896–911. CrossRef
Iaia, J. and Warchall, H., Nonradial solutions of a semilinear elliptic equation in two dimensions. J. Diff. Equ. 119 (1995) 533–558. CrossRef
Kajikiya, R., Norm estimates for radially symmetric solutions of semilinear elliptic equations. Trans. Amer. Math. Soc. 347 (1995) 1163–1199. CrossRef
Kwong, M.K., Uniqueness of positive solutions of $\Delta u -u+u^p=0$ in ${\mathbb{R}} ^n$ . Arch. Rat. Mech. Anal. 105 (1989) 243–266.
Lloyd, D.J.B. and Champneys, A.R., Efficient numerical continuation and stability analysis of spatiotemporal quadratic optical solitons. SIAM J. Sci. Comput. 27 (2005) 759–773. CrossRef
Malomed, B., Drummond, P., He, H., Berntson, A., Anderson, D. and Lisak, M., Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity. Phys. Rev. E 56 (1997) 4725–4735. CrossRef
McLeod, K., Troy, W.C. and Weissler, F.B., Radial solutions of $\Delta u+f(u)=0$ with prescribed number of zeros. J. Diff. Equ. 83 (1990) 368–378. CrossRef
Mizumachi, T., Vortex solitons for 2D focusing nonlinear Schrödinger equation. Diff. Int. Equ. 18 (2005) 431–450.
Moroz, I.M., Penrose, R. and Tod, P., Spherically-symmetric solutions of the Schrödinger-Newton equation. Class. Quant. Grav. 15 (1998) 2733–2742. CrossRef
Steblina, V.V., Kivshar, Y., Lisak, M. and Malomed, B.A., Self-guided beams in diffractive $\chi^{(2)}$ medium: variational approach. Optics Comm. 118 (1995) 345–352. CrossRef
P.L. Sulem and C. Sulem, The nonlinear Schrödinger equation, Self-focusing and wave collapse. AMS, Springer-Verlag (1999).
Towers, I.N., Malomed, B.A. and Wise, F.W., Light bullets in quadratic media with normal dispersion at the second harmonic. Phys. Rev. Lett. 90 (2003) 123902. CrossRef
Weinstein, M.I., Nonlinear Schrödinger equations and sharp interpolation estimates. Comm. Math. Phys. 87 (1983) 567–576. CrossRef