The Ground State of a Gross–Pitaevskii Energy with General Potential in the Thomas–Fermi Limit

Abstract

We study the ground state which minimizes a Gross-Pitaevskii energy with general non-radial trapping potential, under the unit mass constraint, in the ThomasFermi limit where a small parameter ε tends to 0. This ground state plays an important role in the mathematical treatment of recent experiments on the phenomenon of Bose-Einstein condensation, and in the study of various types of solutions of nonhomogeneous defocusing nonlinear Schrödinger equations. Many of these applications require delicate estimates for the behavior of the ground state near the boundary of the condensate, as ε → 0, in the vicinity of which the ground state has irregular behavior in the form of a steep corner layer. In particular, the role of this layer is important in order to detect the presence of vortices in the small density region of the condensate, to understand the superfluid flow around an obstacle, and it also has a leading order contribution in the energy. In contrast to previous approaches, we utilize a perturbation argument to go beyond the clas-

Analysis of an irregular boundary layer behavior for the steady state flow of a Boussinesq fluid

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Energy minimality property of the connecting solution of the Painlevé phase transition model

“…while as x 1 → −∞ it connects to a suitable singular scaling of the hyperbolic tangent kink solution to the Allen-Cahn equation (11). It is worth mentioning that such a solution was previously briefly discussed in [8] (see relation (211) therein). We note in passing that the division trick (6) below, which was not used in [4], may clarify further the interesting link with the Allen-Cahn equation.…”

Theory of light-matter interaction in nematic liquid crystals and the second Painlevé equation

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