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On the effect of a noise on the solutions of the focusing supercritical nonlinear Schrödinger equation
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  • Published: May 2002

On the effect of a noise on the solutions of the focusing supercritical nonlinear Schrödinger equation

  • A. de Bouard1 &
  • A. Debussche2 

Probability Theory and Related Fields volume 123, pages 76–96 (2002)Cite this article

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  • 44 Citations

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Abstract

 We investigate the influence of a random perturbation of white noise type on the finite time blow up of solutions of a focusing supercritical nonlinear Schrödinger equation. We prove that, contrary to the deterministic case, any initial data gives birth to a solution which develops singularities. Moreover, the singularities appear immediately. We use a stochastic generalization of the variance identity and a control argument.

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Authors and Affiliations

  1. CNRS et Université Paris-Sud, UMR 8628, Bât. 425, Université de Paris-Sud, 91405 Orsay, France, , , , , , FR

    A. de Bouard

  2. ENS de Cachan, Antenne de Bretagne, Campus de Ker Lann, Av. R. Schuman, 35170 Bruz, France. e-mail: arnaud.debussche@bretagne.ens-cachan.fr, , , , , , FR

    A. Debussche

Authors
  1. A. de Bouard
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  2. A. Debussche
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Received: 6 February 2001 / Revised version: 18 July 2001 / Published online: 13 May 2002

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de Bouard, A., Debussche, A. On the effect of a noise on the solutions of the focusing supercritical nonlinear Schrödinger equation. Probab Theory Relat Fields 123, 76–96 (2002). https://doi.org/10.1007/s004400100183

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  • Issue Date: May 2002

  • DOI: https://doi.org/10.1007/s004400100183

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Keywords

  • Initial Data
  • White Noise
  • Finite Time
  • Random Perturbation
  • Deterministic Case
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