Well-posedness of the Two-dimensional Nonlinear Schrödinger Equation with Concentrated Nonlinearity

Abstract

We consider a two-dimensional nonlinear Schrödinger equation with concentrated nonlinearity. In both the focusing and defocusing case we prove local well-posedness, i.e., existence and uniqueness of the solution for short times, as well as energy and mass conservation. In addition, we prove that this implies global existence in the defocusing case, irrespective of the power of the nonlinearity, while in the focusing case blowing-up solutions may arise.

39 pages, pdfLaTex. Final version to appear in Ann. I. H. Poincaré - AN

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• Complete ionization for a non-autonomous point interaction model in d = 2
• A general review on the NLS equation with point-concentrated nonlinearity
• Prescribed mass ground states for a doubly nonlinear Schrödinger equation in dimension one
• Microscopic derivation of a Schrödinger equation in dimension one with a nonlinear point interaction
• Well-posedness of the three-dimensional NLS equation with sphere-concentrated nonlinearity