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

Probability Theory and Related Fields

Volumn 123, Issue 1, 2002, Pages 76-96

  De Bouard, A ^a   Debussche, A ^b  

^a UNIVERSITÉ PARIS SUD   (France)

^b UNIVERSITÉ PARIS SACLAY   (France)

Author keywords

Blow up; Nonlinear Schr dinger equation; Stochastic partial differential equation; Variance identity; White noise

Indexed keywords

EID: 0036016202     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s004400100183     Document Type: Article

References (21)

1. Bang, O., Christiansen, P.L., If, F., Rasmussen, K.O., Gaididei, Y.B.: Temperature effects in a nonlinear model of monolayer Scheibe aggregates, Phys. Rev. E 49, 4627-4636 (1994)
2. Bang, O., Christiansen, P.L., If, F., Rasmussen, K.O., Gaididei, Y.B.: White Noise in the Two dimensional Nonlinear Schrödinger Equation, Appl. Anal. 57, 3-15 (1995)
3. Brzezniak, Z.: On stochastic convolution in Banach spaces, and applications, Stochastics Stochastics Rep. 61, 245-295 (1997)
4. Brzezniak, Z., Peszat, S.: Space-time continuous solutions to SPDE's driven by a homogeneous Wiener process, Studia Math. 137, 261-199 (1999)
5. Cazenave, T.: An Introduction to Nonlinear Schrödinger Equations, Textos de Métodos Matématicos 26, Instituto de Matématica-UFRJ Rio de Janeiro, Brazil (1993)
6. Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications, Cambridge University Press (1992)
7. de Bouard, A., Debussche, A.: A stochastic nonlinear Schrödinger equation with multiplicative noise, Comm. Math. Phys. 205, 161-181 (1999)
8. 1, Preprint
9. de Bouard, A., Debussche, A., Tsutsumi, Y.: White noise-driven Korteweg-de Vries equation, J. Funct. Anal. 169, 532-558 (1999)
10. Debussche, A., Dimenza, L.: Numerical simulation of focusing stochastic nonlinear Schrödinger equations, to appear in Physica D
11. Falkovich, G.E., Kolokolov, I., Lebedev, V., Turitsyn, S.K.: Statistics of soliton-bearing systems with additive noise, Phys. Rev. E 63, (2001)
12. Fibich, G., Papanicolaou, G.: Self-focusing in the perturbed and unperturbed nonlinear Schroedinger equation in critical dimension, J. Appl. Math. 60, 183-240 (2000)
13. Ginibre, J., Velo, G.: The global Cauchy problem for the nonlinear Schrödinger equation revisited, Ann. Inst. Henri Poincaré, Analyse Non Linéaire 2, 309-327 (1985)
14. Glassey, R.T.: On the blowing-up of solutions to the Cauchy problem for the nonlinear Schrödinger equation, J. Math. Phys. 18, 1794-1797 (1977)
15. Kato, T.: On Nonlinear Schrödinger equation, Ann. Inst. H. Poincaré, Phys. Theor. 46, 113-129 (1987)
16. Mueller, C.: The Critical Parameter for the Heat Equation with a Noise Term to Blow Up in Finite Time, To appear
17. Mueller, C., Sowers, R.: Blowup for the heat equation with a noise term, Probab. Theory Related Fields 97, 287-320 (1993)
18. Sulem, C., Sulem, P.L.: The Nonlinear Schrödinger Equation, Self-Focusing and Wave Collapse, Appl. Math. Sciences, New York: Springer Verlag (1999)
19. 2 solutions for the nonlinear Schrödinger equation and nonlinear groups, Funkcial. Ekvac. 30, 115-125 (1987)
20. Vlasov, S.N., Petrishchev, V.A., Talanov, V.I.: Average description of wave beams in linear and nonlinear media (The method of moments), Radiophys. Quantum Electron. 14, 1062-1070 (1974)
21. Zakharov, V.E.: Collapse of Langmuir waves, Sov. Phys. JETP 35, 908-914 (1972)