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Finite dimensional behavior for weakly damped driven Schrödinger equations. (English) Zbl 0659.35019
We study the long time behavior of nonlinear Schrödinger equations with a zero order dissipation when they are driven by an external force. We show that this behavior is described by an attractor which captures all the trajectories. One of our main results concerns the estimate of the uniform Lyapunov exponents on this attractor, which allows us to prove its finite dimensional character.

MSC:
35G20 Nonlinear higher-order PDEs
35B40 Asymptotic behavior of solutions to PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
76A99 Foundations, constitutive equations, rheology, hydrodynamical models of non-fluid phenomena
78A60 Lasers, masers, optical bistability, nonlinear optics
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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References:
[1] Blow, K. J.; Doran, N. J., Global and local chaos in the pumped nonlinear Schrödinger equation, Physical Review Letters, Vol. 52, No. 7, 526-539, (1984)
[2] Bourbaki, N., Espaces vectoriels topologiques, (1981), Masson Paris · Zbl 0482.46001
[3] Constantin, P.; Foias, C.; Temam, R., Attractors representing turbulent flows, Memoirs of A.M.S., Vol. 53, No. 314, (1985) · Zbl 0567.35070
[4] Courant, R.; Hilbert, D., Methods of Mathematical Physics, Vol. I, 34-36, (1966), Interscience New-York
[5] Douady, A.; Oesterlé, J., Dimension de Hausdorff des attracteurs, C.R. Acad. Sci. Paris, T. 290, Series A, 1135-1138, (1980) · Zbl 0443.58016
[6] Ghidaglia, J. M.; Héron, B., Dimension of the attractors associated to the Ginzburg-Landau partial differential equation, Physica, Vol. 28D, 282-304, (1987) · Zbl 0623.58049
[7] Ghidaglia, J. M.; Temam, R., Attractors for damped nonlinear hyperbolic equations, J. Math. Pures Appl., T. 66, 273-319, (1987) · 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). · 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). · Zbl 0688.58027
[10] Glassey, R. T., On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., Vol. 18, No. 9, 1794-1797, (1977) · Zbl 0372.35009
[11] Lions, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires, (1969), Dunod Paris · Zbl 0189.40603
[12] Lions, J. L.; Magenes, E., Nonhomogeneous boundary value problems and applications, (1972), Springer Berlin, (translated from Dunod, Paris, 1968) · Zbl 0227.35001
[13] Maldelbrot, B., Fractals: form, chance and dimension, (1977), Freeman San Francisco
[14] Nozaki, K.; Bekki, N., Low-dimensional chaos in a driven damped nonlinear Schrödinger equation, Physica, Vol. 21D, 381-393, (1986) · Zbl 0607.35017
[15] Segal, I., Ann. Math., Vol. 78, 339-364, (1963)
[16] Temam, R., Infinite dimensional dynamical systems in mechanics and physics, (1988), Springer-Verlag New York · Zbl 0662.35001
[17] Tsutsumi, M., Non existence of global solutions to the Cauchy problem for the damped nonlinear Schrödinger equations, S.I.A.M. J. Math. Anal., Vol. 15, 357-366, (1984) · Zbl 0539.35022
[18] Zakharov, V. E.; Shabat, A. B., Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, J.E.P.P., Vol. 34, 62-69, (1972)
[19] Ghidaglia, J. M., Comportement de dimension finie pour LES équations de Schrödinger non linéaires faiblement amorties, C.R. Acad. Sci. Paris, T. 305, Series I, 291-294, (1987) · Zbl 0638.35020
[20] J. M. Ghidaglia, Weekly Damped Forced Korteweg- de Vries Equations Behave as a Finite Dimensional Dynamical System in the Long Time, J. Diff. Equ. (in the Press). · Zbl 0668.35084
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