Finite dimensional behavior for weakly damped driven Schrödinger equations
Annales de l'I.H.P. Analyse non linéaire, Volume 5 (1988) no. 4, p. 365-405
@article{AIHPC_1988__5_4_365_0,
     author = {Ghidaglia, Jean-Michel},
     title = {Finite dimensional behavior for weakly damped driven Schr\"odinger equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Gauthier-Villars},
     volume = {5},
     number = {4},
     year = {1988},
     pages = {365-405},
     zbl = {0659.35019},
     mrnumber = {963105},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_1988__5_4_365_0}
}
Ghidaglia, Jean-Michel. Finite dimensional behavior for weakly damped driven Schrödinger equations. Annales de l'I.H.P. Analyse non linéaire, Volume 5 (1988) no. 4, pp. 365-405. http://www.numdam.org/item/AIHPC_1988__5_4_365_0/

[1] K.J. Blow and N.J. Doran, Global and Local Chaos in the Pumped Nonlinear Schrödinger Equation, Physical Review Letters, Vol. 52, No. 7, 1984, pp. 526-539.

[2] N. Bourbaki, Espaces vectoriels topologiques, Masson, Paris, 1981. | MR 633754 | Zbl 0482.46001

[3] P. Constantin, C. Foias and R. Temam, Attractors Representing Turbulent Flows, Memoirs of A.M.S., Vol. 53, No. 314, 1985. | MR 776345 | Zbl 0567.35070

[4] R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. I, 1966, pp. 34- 36. Interscience, New-York. | MR 65391 | Zbl 0051.28802

[5] A. Douady and J. Oesterlé, Dimension de Hausdorff des attracteurs. C.R. Acad. Sci. Paris, T. 290, Series A, 1980, pp. 1135-1138. | MR 585918 | Zbl 0443.58016

[6] J.M. Ghidaglia and B. Héron, Dimension of the Attractors Associated to the Ginzburg-Landau Partial Differential Equation, Physica, Vol. 28D, 1987, pp. 282- 304.. | MR 914451 | Zbl 0623.58049

[7] J.M. Ghidaglia and R. Temam, Attractors for Damped Nonlinear Hyperbolic Equations, J. Math. Pures Appl., T. 66, 1987, pp. 273-319. | MR 913856 | Zbl 0572.35071

[8] J.M. Ghidaglia and R. Temam, Regularity of the Solutions of Second Order Evolution Equations and Their Attractors, Annali della scuola Normale sup. di Pisa (in the Press). | Numdam | Zbl 0666.35062

[9] J.M. Ghidaglia and R. Temam, Periodic Dynamical System with Application to Sine Gordon Equations: Estimates on the Fractal Dimension of the Universal Attractor, Proceedings of the Boulder Conference, B. NICOLAENKO Ed., Contemporary Math., A.M.S., Providence (to appear). | MR 1034498 | Zbl 0688.58027

[10] R.T. Glassey, On the Blowing up of Solutions to the Cauchy Problem for Nonlinear Schrödinger Equations, J. Math. Phys., Vol. 18, No. 9, 1977, pp. 1794-1797. | MR 460850 | Zbl 0372.35009

[11] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969. | MR 259693 | Zbl 0189.40603

[12] J.L. Lions and E. Magenes, Nonhomogeneous Boundary value Problems and Applications, Springer, Berlin, 1972 (translated from Dunod, Paris, 1968).

[13] B. Maldelbrot, Fractals: Form, Chance and Dimension, Freeman, San Francisco, 1977. | MR 471493 | Zbl 0376.28020

[14] K. Nozaki and N. Bekki, Low-Dimensional Chaos in a Driven Damped Nonlinear Schrödinger Equation, Physica, Vol. 21D, 1986, pp. 381-393. | MR 862265 | Zbl 0607.35017

[15] I. Segal, Nonlinear Semi-Groups, Ann. Math., Vol. 78, 1963, pp. 339-364. | MR 152908 | Zbl 0204.16004

[16] R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, New York, 1988. | MR 953967 | Zbl 0662.35001

[17] M. Tsutsumi, Non Existence of Global Solutions to the Cauchy Problem for the Damped Nonlinear Schrödinger Equations, S.I.A.M. J. Math. Anal., Vol. 15, 1984, pp. 357-366. | MR 731873 | Zbl 0539.35022

[18] V.E. Zakharov and A.B. Shabat, Exact Theory of Two-Dimensional Self-Focusing and One-Dimensional Self-Modulation of Waves in Nonlinear Media, J.E.P.P., Vol. 34, 1972, pp. 62-69. | MR 406174

[19] J.M. Ghidaglia, Comportement de dimension finie pour les équations de Schrödinger non linéaires faiblement amorties, C.R. Acad. Sci. Paris, Series I, T. 305, 1987, pp. 291-294. | MR 910362 | Zbl 0638.35020

[20] J.M. Ghidaglia, Weekly Damped Forced Kortewegde Vries Equations Behave as a Finite Dimensional Dynamical System in the Long Time, J. Diff. Equ. (in the Press). | Zbl 0668.35084