Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-26T11:03:30.629Z Has data issue: false hasContentIssue false
  • English
  • Français

Large deviations and support results fornonlinear Schrödinger equations with additive noise andapplications

Published online by Cambridge University Press:  15 November 2005

Éric Gautier
Éric Gautier*
Affiliation:
CREST-INSEE, URA D2200, 3 avenue Pierre Larousse, 92240 Malakoff, France. IRMAR, UMR 6625, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France; eric.gautier@bretagne.ens-cachan.fr
Get access

Abstract

Sample path large deviations for the laws of the solutions of stochastic nonlinear Schrödinger equations when the noise converges to zero are presented. The noise is a complex additive Gaussian noise. It is white in time and colored in space. The solutions may be global or blow-up in finite time, the two cases are distinguished. The results are stated in trajectory spaces endowed with topologies analogue to projective limit topologies. In this setting, the support of the law of the solution is also characterized. As a consequence, results on the law of the blow-up time and asymptotics when the noise converges to zero are obtained. An application to the transmission of solitary waves in fiber optics is also given.

Keywords

Large deviationsstochastic partial differential equationsnonlinear Schrödinger equationswhite noiseprojective limitsupport theoremblow-upsolitary waves.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 9 , February 2005 , pp. 74 - 97
Copyright
© EDP Sciences, SMAI, 2005

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

Azencott, R., Grandes déviations et applications, in École d'été de Probabilité de Saint-Flour, P.L. Hennequin Ed. Springer-Verlag, Berlin. Lect. Notes Math. 774 (1980) 1–176. CrossRef
A. Badrikian and S. Chevet, Mesures cylindriques, espaces de Wiener et fonctions aléatoires Gaussiennes. Springer-Verlag, Berlin. Lect. Notes Math. 379 (1974).
Berezansky, Y.M., Sheftel, Z.G. and Us, G.F., Functional Analysis, Vol. 1. Oper. Theor. Adv. Appl. 85 (1997) 125–134.
G. Buttazzo, M. Giaquinta and S. Hildebrandt, One-dimensional Variational Problems. Oxford University Press, Oxford. Oxford Lect. Ser. Math. Appl. 15 (1998).
T. Cazenave, An Introduction to Nonlinear Schrödinger Equations. Instituto de Matématica-UFRJ Rio de Janeiro, Brazil. Textos de Métodos Matématicos 26 (1993).
G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions. Cambridge University Press: Cambridge, England. Encyclopedia Math. Appl. (1992).
de Bouard, A. and Debussche, A., The Stochastic Nonlinear Schrödinger Equation in H1. Stochastic Anal. Appl. 21 (2003) 97–126. CrossRef
de Bouard, A. and Debussche, A., On the effect of a noise on the solutions of the focusing supercritical nonlinear Schrödinger equation. Probab. Theory Relat. Fields 123 (2002) 76–96. CrossRef
de Bouard, A. and Debussche, A., Finite time blow-up in the additive supercritical nonlinear Schrödinger equation: the real noise case. Contemp. Math. 301 (2002) 183–194. CrossRef
Debussche, A. and Di Menza, L., Numerical simulation of focusing stochastic nonlinear Schrödinger equations. Phys. D 162 (2002) 131–154. CrossRef
Derevyanko, S.A., Turitsyn, S.K. and Yakusev, D.A., Non-gaussian statistics of an optical soliton in the presence of amplified spontaneaous emission. Optics Lett. 28 (2003) 2097–2099. CrossRef
J.D. Deuschel and D.W. Stroock, Large Deviations. Academic Press, New York. Pure Appl. Math. (1986).
A. Dembo and O. Zeitouni, Large deviation techniques and applications (2nd edition). Springer-Verlag, New York. Appl. Math. 38 (1998).
Drummond, P.D. and Corney, J.F., Quantum noise in optical fibers. II. Raman jitter in soliton communications. J. Opt. Soc. Am. B 18 (2001) 153–161. CrossRef
L.C. Evans, Partial Differential Equations. American Mathematical Society, Providence, Rhode Island, Grad. Stud. in Math. 119 (1998).
Falkovich, G.E., Kolokolov, I., Lebedev, V. and Turitsyn, S.K., Statistics of soliton-bearing systems with additive noise. Phys. Rev. E 63 (2001) 025601(R). CrossRef
Falkovich, G., Kolokolov, I., Lebedev, V., Mezentsev, V. and Turitsyn, S.K., Non-Gaussian error probability in optical soliton transmission. Physica D 195 (2004) 1–28. CrossRef
É. Gautier, Uniform large deviations for the nonlinear Schrödinger equation with multiplicative noise. Preprint IRMAR, Rennes (2004). Submitted for publication.
T. Kato, On Nonlinear Schrödinger Equation. Ann. Inst. H. Poincaré, Phys. Théor. 46 (1987) 113–129.
V. Konotop and L. Vázquez, Nonlinear random waves. World Scientific Publishing Co., Inc.: River Edge, New Jersey (1994).
Moore, R.O., Biondini, G. and Kath, W.L., Importance sampling for noise-induced amplitude and timing jitter in soliton transmission systems. Optics Lett. 28 (2003) 105–107. CrossRef
C. Sulem and P.L. Sulem, The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse. Springer-Verlag, New York, Appl. Math. Sci. (1999).
J.B. Walsh, An introduction to stochastic partial differential equations, in École d'été de Probabilité de Saint-Flour, P.L. Hennequin Ed. Springer-Verlag, Berlin, Lect. Notes Math. 1180 (1986) 265–439.