In the present paper, we study the following scaled nonlinear Schrödinger equation (NLS) in one space dimension:(Formula Presented.)(Formula Presented.) This equation represents a nonlinear Schrödinger equation with a spatially concentrated nonlinearity. We show that in the limit (Formula Presented.) the weak (integral) dynamics converges in (Formula Presented.) to the weak dynamics of the NLS with point-concentrated nonlinearity: (Formula Presented.) where Hα is the Laplacian with the nonlinear boundary condition at the origin (Formula Presented.) and (Formula Presented.). The convergence occurs for every (Formula Presented.) if V ≥ 0 and for every (Formula Presented.) otherwise. The same result holds true for a nonlinearity with an arbitrary number N of concentration points.

The NLS Equation in Dimension One with Spatially Concentrated Nonlinearities: the Pointlike Limit

CACCIAPUOTI, CLAUDIO;
2014-01-01

Abstract

In the present paper, we study the following scaled nonlinear Schrödinger equation (NLS) in one space dimension:(Formula Presented.)(Formula Presented.) This equation represents a nonlinear Schrödinger equation with a spatially concentrated nonlinearity. We show that in the limit (Formula Presented.) the weak (integral) dynamics converges in (Formula Presented.) to the weak dynamics of the NLS with point-concentrated nonlinearity: (Formula Presented.) where Hα is the Laplacian with the nonlinear boundary condition at the origin (Formula Presented.) and (Formula Presented.). The convergence occurs for every (Formula Presented.) if V ≥ 0 and for every (Formula Presented.) otherwise. The same result holds true for a nonlinearity with an arbitrary number N of concentration points.
2014
nonlinear delta interactions; Nonlinear Schrödinger equation; zero-range limit of concentrated nonlinearities
Cacciapuoti, Claudio; Finco, D.; Noja, D.; Teta, A.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11383/1970720
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