The Annals of Probability

Blow-up for the stochastic nonlinear Schrödinger equation with multiplicative noise

Arnaud Debussche and Anne de Bouard

Full-text: Open access

Abstract

We study the influence of a multiplicative Gaussian noise, white in time and correlated in space, on the blow-up phenomenon in the supercritical nonlinear Schrödinger equation. We prove that any sufficiently regular and localized deterministic initial data gives rise to a solution which blows up in arbitrarily small time with a positive probability.

Article information

Source
Ann. Probab., Volume 33, Number 3 (2005), 1078-1110.

Dates
First available in Project Euclid: 6 May 2005

Permanent link to this document
https://projecteuclid.org/euclid.aop/1115386719

Digital Object Identifier
doi:10.1214/009117904000000964

Mathematical Reviews number (MathSciNet)
MR2135313

Zentralblatt MATH identifier
1068.35191

Subjects
Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10] 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 76B35 60H30: Applications of stochastic analysis (to PDE, etc.) 60J60: Diffusion processes [See also 58J65]

Keywords
Nonlinear Schrödinger equations stochastic partial differential equations white noise blow-up variance identity support theorem

Citation

de Bouard, Anne; Debussche, Arnaud. Blow-up for the stochastic nonlinear Schrödinger equation with multiplicative noise. Ann. Probab. 33 (2005), no. 3, 1078--1110. doi:10.1214/009117904000000964. https://projecteuclid.org/euclid.aop/1115386719


Export citation

References

  • Bang, O., Christiansen, P. L., If, F., Rasmussen, K. O. and Gaididei, Y. B. (1994). Temperature effects in a nonlinear model of monolayer Scheibe aggregates. Phys. Rev. E 49 4627--4636.
  • Bang, O., Christiansen, P. L., If, F., Rasmussen K. O. and Gaididei, Y. B. (1995). White noise in the two-dimensional nonlinear Schrödinger equation. Appl. Anal. 57 3--15.
  • Brzezniak, Z. (1997). On stochastic convolution in Banach spaces, and applications. Stochastics Stochastics Rep. 61 245--295.
  • Brzezniak, Z. and Peszat, S. (1999). Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process. Studia Math. 137 261--299.
  • Cazenave, T. (1993). An Introduction to Nonlinear Schrödinger Equations. Textos de Métodos Matématicos 26, Instituto de Matématica-UFRJ Rio de Janeiro, Brazil.
  • Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Cambridge Univ. Press.
  • de Bouard, A. and Debussche, A. (1999). A stochastic nonlinear Schrödinger equation with multiplicative noise. Comm. Math. Phys. 205 161--181.
  • de Bouard, A. and Debussche, A. (2002). On the effect of a noise on the solutions of the focusing supercritical nonlinear Schrödinger equation. Probab. Theory Related Fields 123 76--96.
  • de Bouard, A. and Debussche, A. (2002). Finite time blow-up in the additive supercritical stochastic nonlinear Schrödinger equation: The real noise case. Contemp. Math. 301 183--194.
  • de Bouard, A. and Debussche, A. (2003). The stochastic nonlinear Schrödinger equation in $H^1$. Stochastic Anal. Appl. 21 97--126.
  • de Bouard, A. and Debussche, A. (2004). A semi-discrete scheme for the stochastic nonlinear Schrödinger equation. Numer. Math. 96 733--770.
  • Debussche, A. and Di Menza, L. (2002). Numerical simulation of focusing stochastic nonlinear Schrödinger equations. Phys. D 162 131--154.
  • Falkovich, G. E., Kolokolov, I., Lebedev, V. and Turitsyn, S. K. (2001). Statistics of soliton-bearing systems with additive noise. Phys. Rev. E 63 025601(R).
  • Garnier, J. (1998). Asymptotic transmission of solitons through random media. SIAM J. Appl. Math. 58 1969--1995.
  • Glassey, R. T. (1977). On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation. J. Math. Phys. 18 1794--1797.
  • Gyöngy, I. and Krylov, N. V. (1996). Existence of strong solutions for Itô's stochastic equations via approximations. Probab. Theory Related Fields 105 143--158.
  • Kato, T. (1987). On nonlinear Schrödinger equation. Ann. Inst. H. Poincaré Phys. Théor. 46 113--129.
  • Konotop, V. and Vázquez, L. (1994). Nonlinear Random Waves. World Scientific, River Edge, NJ.
  • Millet, A. and Sanz-Solé, M. (1994). A simple proof of the support theorem for diffusion processes. Séminaire de Probabilités XXVIII. Lecture Notes in Math. 1583 36--48. Springer, Berlin.
  • Sulem, C. and Sulem, P. L. (1999). The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse. Springer, New York.
  • Ueda, T. and Kath, W. L. (1992). Dynamics of optical pulses in randomly birefrengent fibers. Phys. D 55 166--181.
  • Zakharov, V. E. (1972). Collapse of Langmuir waves. Soviet Phys. JETP 35 908--914.