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We study the long-time behavior of the solutions to a nonlinear damped driven Schrödinger type equation with quadratic potential on a strip. We prove that this behavior is described by a regular compact global attractor with finite fractal dimension.2000 Mathematics Subject Classification. Primary: 35L05, 35Q55; Secondary: 76B03.

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In this article, we present, throughout two basic models of damped nonlinear Schrödinger (NLS)-type equations, a new idea to bound from above the fractal dimension of the global attractors for NLS-type equations. This could answer the following open issue: consider, for instance, the classical one-dimensional cubic nonlinear Schrödinger equation u t + iu xx + i|u| 2 u + u = , ∈ L 2 (R). "How can we bound the fractal dimension of the associate global attractor without the need to assume that the external forcing term f has some decay at infinity

We study the long time behaviour of the solutions for a class of nonlinear damped fractional Schrödinger type equation with anisotropic dispersion and in presence of a quadratic potential in a two dimensional unbounded domain. We prove that this behaviour is characterized by the existence of regular compact global attractor with finite fractal dimension.