We consider a two-dimensional nonlinear Schr\"odinger 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.
Well-posedness of the two-dimensional nonlinear Schrödinger equation with concentrated nonlinearity / Carlone, Raffaele; Correggi, Michele; Tentarelli, Lorenzo. - In: ANNALES DE L INSTITUT HENRI POINCARÉ. ANALYSE NON LINÉAIRE. - ISSN 0294-1449. - (2018). [10.1016/j.anihpc.2018.05.003]
Well-posedness of the two-dimensional nonlinear Schrödinger equation with concentrated nonlinearity
Correggi, Michele
;Tentarelli, Lorenzo
2018
Abstract
We consider a two-dimensional nonlinear Schr\"odinger 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.| File | Dimensione | Formato | |
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