We consider a scaling limit of a nonlinear Schrödinger equation (NLS) with a nonlocal nonlinearity showing that it reproduces in the limit of cutoff removal a NLS equation with nonlinearity concentrated at a point. The regularized dynamics is described by the equation. i∂∂tψε(t)=-δψε(t)+g(ε,μ,|(ρε,ψε(t))|2μ)(ρε,ψε(t))ρε where ρε→δ0 weakly and the function g embodies the nonlinearity and the scaling and has to be fine tuned in order to have a nontrivial limit dynamics. The limit dynamics is a nonlinear version of point interaction in dimension three and it has been previously studied in several papers as regards the well-posedness, blow-up and asymptotic properties of solutions. Our result is the first justification of the model as the point limit of a regularized dynamics.

### The point-like limit for a NLS equation with concentrated nonlinearity in dimension three

#### Abstract

We consider a scaling limit of a nonlinear Schrödinger equation (NLS) with a nonlocal nonlinearity showing that it reproduces in the limit of cutoff removal a NLS equation with nonlinearity concentrated at a point. The regularized dynamics is described by the equation. i∂∂tψε(t)=-δψε(t)+g(ε,μ,|(ρε,ψε(t))|2μ)(ρε,ψε(t))ρε where ρε→δ0 weakly and the function g embodies the nonlinearity and the scaling and has to be fine tuned in order to have a nontrivial limit dynamics. The limit dynamics is a nonlinear version of point interaction in dimension three and it has been previously studied in several papers as regards the well-posedness, blow-up and asymptotic properties of solutions. Our result is the first justification of the model as the point limit of a regularized dynamics.
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2017
http://www.sciencedirect.com/science/article/pii/S002212361730157X
Nonlinear delta interactions; Nonlinear Schrödinger equation; Zero-range limit of concentrated nonlinearities; Analysis
Cacciapuoti, Claudio; Finco, Domenico; Noja, Diego; Teta, Alessandro
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11383/2063240`
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