Dynamics of a Gross-Pitaevskii Equation with Phenomenological Damping

We study the dynamical behavior of solutions of an n-dimensional nonlinear Schr？dinger equation with potential and linear derivative terms under the presence of phenomenological damping. This equation is a general version of the dissipative Gross-Pitaevskii equation including terms with first-order derivatives in the spatial coordinates which allow for rotational contributions. We obtain conditions for the existence of a global attractor and find bounds for its dimension. 1. Introduction Nonlinear Schr？dinger equations have enjoyed a considerable amount of attention during several decades due to their frequent appearance in the modeling of interesting physical phenomena in many different areas (e.g., optics, fluid mechanics, condensed matter, etc.). Additionally, the rigorous mathematical treatment of these equations has produced a great deal of new insights and techniques accompanied by a voluminous bibliography. We refer the reader to excellent sources as [1–4] for a detailed account of these aspects. Here, for complex coefficients and , our purpose is to study the long-term dynamics of a general nonlinear Schr？dinger evolution equation with potential of the form where and are complex-valued functions and is a complex vector-valued map, all defined on a bounded open domain (for ) with regular boundary . We prove the existence of a global attractor for (1), under Dirichlet and -periodic boundary conditions, assuming and some additional requirements on and for which existence and uniqueness of solutions are guaranteed. Our choice of signs for the terms in this equation was made such that the hypotheses needed for our analysis make, if necessary, the imaginary parts of constant and functional coefficients positive. Particular instances of (1) have been extensively considered in the literature, specially in two and three dimensions. In fact, when the coefficients , and and the function are real and assuming , this is the celebrated time-dependent Gross-Pitaevskii equation (GP) employed to model nonlinear behavior in several physical systems. Of special relevance is the mean-field description of Bose-Einstein condensation achieved in dilute atomic gases, a state of matter which exhibits peculiar phenomena characteristic of the superfluid nature of the system. If for the previous situation in three dimensions we choose , where is an angular velocity vector, we recover the GP equation with angular momentum rotational term. Indeed, the right-hand side of (1) becomes where is precisely the quantum mechanical angular momentum operator. This equation has been