Finite dimensional global attractor for a semi-discrete fractional nonlinear Schrödinger equation
Résumé
We consider a semi-discrete in time Crank-Nicolson scheme to discretise a weakly damped forced nonlinear fractional Schrödinger equation u t − i(−∆) α u + i|u| 2 u + γu = f for α ∈ (1 2 , 1) considered in the the whole space R. We prove that such semi-discrete equation provides a discrete infinite dimensional dynamical in H α (R) that possesses a global attractor in H α (R). We show also that if the external force is in a suitable weighted Lebesgue space then this global attractor has a finite fractal dimension.
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