Abstract
We study the dynamical behavior of solutions of ann-dimensional nonlinear Schrödinger 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.
Highlights
We study the dynamical behavior of solutions of an n-dimensional nonlinear Schrodinger equation with potential and linear derivative terms under the presence of phenomenological damping
Nonlinear Schrodinger 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
This should be compared with the role played by this term in proving finite time blowup for the nondissipative focusing case in [10], alluded in Remark 9
Summary
Nonlinear Schrodinger 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.). Since the arguments needed for our analysis work in the fully complex coefficients situation, we adopt this broader setting and, in addition, do not necessarily assume that the right-hand side of (1) has the particular form of an angular momentum contribution This situation corresponds to the case where the ratio between imaginary and real parts of α, β, f(x), and V(x) in (3) is not a fixed constant γ. We point out that numerical studies in [27] show that cubic or quintic damping terms prevent blowup of solutions to focusing nonlinear Schrodinger equations without “rotational” term, while just linear damping is not enough below a certain threshold This agrees with Theorem 8 on existence of attractors (that covers such situation if one assumes an arbitrarily small amount of linear damping) and provides some justification for the hypotheses under which this result is proved. It is important to notice that the estimates use only the L2-norm of u0, which exists by (H5)
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