Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-28T04:07:22.718Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical study of self-focusing solutions to the Schrödinger-Debye system

Published online by Cambridge University Press:  15 April 2002

Christophe Besse  and
Brigitte Bidégaray
Christophe Besse
Affiliation:
Laboratoire MIP, CNRS UMR 5640, Université Paul Sabatier, 118 route de Narbonne, Toulouse Cedex 4, France. (besse@mip.ups-tlse.fr; bidegara@mip.ups-tlse.fr)
Brigitte Bidégaray
Affiliation:
Laboratoire MIP, CNRS UMR 5640, Université Paul Sabatier, 118 route de Narbonne, Toulouse Cedex 4, France. (besse@mip.ups-tlse.fr; bidegara@mip.ups-tlse.fr)
Get access

Abstract

In this article we implement different numerical schemes to simulate the Schrödinger-Debye equations that occur in nonlinear optics. Since the existence of blow-up solutions is an open problem, we try to compute such solutions. The convergence of the methods is proved and simulations seem indeed to show that for at least small delays self-focusing solutions may exist.

Keywords

Nonlinear opticsSchrödinger-like equationsrelaxation methodsplit-step methodself-focusing.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 1 , January 2001 , pp. 35 - 55
Copyright
© EDP Sciences, SMAI, 2001

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

G.D. Akrivis, V.A. Dougalis, O.A. Karakashian and W.R. McKinney, Numerical approximation of blow-up of radially symmetric solutions of the nonlinear Schrödinger equation (1997). Preprint.
C. Besse, Schéma de relaxation pour l'équation de Schrödinger non linéaire et les systèmes de Davey et Stewartson. C. R. Acad. Sci., Sér. I 326 (1998) 1427-1432.
C. Besse, Analyse numérique des systèmes de Davey-Stewartson. Ph.D. thesis, University of Bordeaux I, France (1998).
C. Besse, B. Bidégaray and S. Descombes, Accuracy of the split-step schemes for the Nonlinear Schrödinger Equation. (In preparation).
Bidégaray, B., On the Cauchy problem for systems occurring in nonlinear optics. Adv. Differential Equations 3 (1998) 473-496.
Bidégaray, B., The Cauchy problem for Schrödinger-Debye equations. Math. Models Methods Appl. Sci. 10 (2000) 307-315.
J.L. Bona, V.A. Dougalis, O.A. Karakashian and W.R. McKinney, Conservative, high-order numerical schemes for the generalized Korteweg-de Vries equation. Philos. Trans. Roy. Soc. London, Ser. A 351 (1995) 107-164.
T. Cazenave, An introduction to nonlinear Schrödinger equations. Textos de métodos matemáticos 26, Rio de Janeiro (1990).
T. Cazenave, Blow-up and Scattering in the nonlinear Schrödinger equation. Textos de métodos matemáticos 30, Rio de Janeiro (1994).
Colin, T. and Fabrie, P., Semidiscretization in time for nonlinear Schrödinger-waves equations. Discrete Contin. Dynam. Systems 4 (1998) 671-690.
Delfour, M., Fortin, M. and Payre, G., Finite-difference solutions of a nonlinear Schrödinger equation. J. Comput. Phys. 44 (1981) 277-288. CrossRef
B.O. Dia and M. Schatzman, Estimations sur la formule de Strang. C. R. Acad. Sci. Paris, Sér. I 320 (1995) 775-779.
L. Di Menza, Approximations numériques d'équations de Schrödinger non linéaires et de modèles associés. Ph.D. thesis, University of Bordeaux I, France (1995).
P. Donnat, Quelques contributions mathématiques en optique non linéaire. Ph.D. thesis, École Polytechnique, France (1994).
Fibich, G. and Papanicolaou, G.C., Self-focusing in the perturbed and unperturbed nonlinear Schrödinger equation in critical dimension. SIAM J. Appl. Math. 60 (2000) 183-240. CrossRef
Glassey, R.T., Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension. Math. Comput. 58 (1992) 83-102. CrossRef
A.C. Newell and J.V. Moloney, Nonlinear Optics. Addison-Wesley (1992).
J.M. Sanz-Serna Methods for the Numerical Solution of the Nonlinear Schrödinger Equation. Math. Comput. 43 (1984) 21-27
Y. R. Shen, The Principles of Nonlinear Optics. Wiley, New York (1984).
G. Strang On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5 (1968) 506-517.
Sulem, C., Sulem, P.L. and Patera, A., Numerical Simulation of Singular Solutions to the Two-Dimensional Cubic Schrödinger Equation. Commun. Pure Appl. Math. 37 (1984) 755-778. CrossRef
Weideman, J.A.C. and Herbst, B.M., Split-step methods for the solution of the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 23 (1986) 485-507. CrossRef