Explicit upper and lower bounds on the number of degrees of freedom for damped and driven cubic Schrödinger equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 23 (1989) no. 3, p. 433-443
@article{M2AN_1989__23_3_433_0,
     author = {Ghidaglia, Jean-Michel},
     title = {Explicit upper and lower bounds on the number of degrees of freedom for damped and driven cubic Schr\"odinger equations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {Dunod},
     volume = {23},
     number = {3},
     year = {1989},
     pages = {433-443},
     zbl = {0688.35084},
     mrnumber = {1014484},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_1989__23_3_433_0}
}
Ghidaglia, J. M. Explicit upper and lower bounds on the number of degrees of freedom for damped and driven cubic Schrödinger equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 23 (1989) no. 3, pp. 433-443. http://www.numdam.org/item/M2AN_1989__23_3_433_0/

[1] J. E. Billoti and J. P. La Salle, Dissipative periodic processes, Bull. Amer. Math. Soc. 77 (1971) 1082-1088. | MR 284682 | Zbl 0274.34061

[2] K. J. Blow and N. J. Doran, Global and local chaos in the pumped nonlinear Schrödinger equation, Physical Review Letters 52 (1984) 526-539.

[3] P. Constantin, C. Foias and R. Temam , Attractors representing turbulent flows, Memoirs of A.M.S. 53 (1985) n° 314. | MR 776345 | Zbl 0567.35070

[4] J. M. Ghidaglia, Comportement de dimension finie pour les équations de Schrödinger non linéaires faiblement amorties, C.R. Acad. Sci. Paris, t. 305, Série I (1987) 291-294. | MR 910362 | Zbl 0638.35020

[5] J. M. Ghidaglia, Finite dimensional behavior for weakly damped driven Schrodinger equations, Ann. Inst. Henri Poincaré, Analyse Non Linéaire 5 (1988) 365-405. | Numdam | MR 963105 | Zbl 0659.35019

[6] J. M. Ghidaglia, Weakly damped forced Korteweg-de Vries equations behave as a finite dimensional dynamical System in the long time, J. Diff. Equ. 74 (1988) 369-390. | MR 952903 | Zbl 0668.35084

[7] J. M. Ghidaglia and B. Héron, Dimension of the attractors associated to the Ginzburg-Landau partial differential equation, Physica 28D (1987) 282-304. | MR 914451 | Zbl 0623.58049

[8] J. M. Ghidaglia and R. Temam Attractors for damped nonlinear hyperbolic equations, J. Math. Pures Appl. 66 (1987) 282-304. | MR 913856

[9] K. Nozaki and N. Bekki, Low-dimensional chaos in a driven damped nonlinear Schrödinger equation, Physica 21D (1986) 381-393. | MR 862265 | Zbl 0607.35017

[10] N. Levinson , Transformation theory of nonlniear differential equations of the second order, Annals of Math. 45 (1944) 723-737. | MR 11505 | Zbl 0061.18910

V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP 34 (1972) 62-39. | MR 406174