Pullback dynamics and statistical solutions for semi-dissipative nonlinear Schrödinger equations

Abstract

This article investigates the pullback dynamics and statistical solutions for the semi-dissipative nonlinear Schrödinger equations describing the model of Bose–Einstein condensate with an impurity. The principal difficulty comes from the semi-dissipative property described by the fact that the first equation is dissipative and the second one is conservative. First, we prove that the generated evolution process of the solution mappings possesses a pullback attractor, which pullback attracts any bounded subset of the phase space with respect to a certain mixed topology. Second, we establish that the evolution process has a family of invariant Borel probability measures, and these measures satisfy the Liouville type equation. Finally, we verify the equivalence between the statistical solutions and invariant measures. The obtained results reveal that the Boson wave function plays the dominant role while the impurity wave function plays the subordinate role in the evolution dynamics of the addressed model.