Vortices in Bose—Einstein Condensates

Volume 67 of the series Progress in Nonlinear Differential Equations and Their Applications pp 19-28

The Mathematical Setting: A Survey of the Main Theorems

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This book contains results in three directions: the Thomas-Fermi regime or small ε problem, where there is a bounded number of vortices in the system (Chapters 3, 4, 6); the fast-rotation regime, which displays a vortex lattice (Chapter 5); and the experiment of a superfluid flow around an obstacle (Chapter 7). The tools and techniques needed to address these problems are very different: energy expansion using a small parameter for the Thomas-Fermi regime; double-scale convergence, homogenization techniques, and Fock-Bargmann space for the fast-rotating regime; energy estimates and nondegeneracy of a solution for the superfluid flow. The main mathematical results are summarized in the present chapter.