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arXiv · 1511.06731

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

Claudio Cacciapuoti·Domenico Finco·Diego Noja·Alessandro Teta
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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 \begin{equation*} i\frac{\partial }{\partial t} ψ^\varepsilon(t)= -Δψ^\varepsilon(t) + g(\varepsilon,μ,|(ρ^\varepsilon,ψ^\varepsilon(t))|^{2μ}) (ρ^\varepsilon,ψ^\varepsilon(t)) ρ^\varepsilon \end{equation*} where $ρ^{\varepsilon} \to δ_0$ 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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Claudio Cacciapuoti, Domenico Finco, Diego Noja, Alessandro Teta. 2017-06-29. The point-like limit for a NLS equation with concentrated nonlinearity in dimension three. https://doi.org/10.1016/j.jfa.2017.04.011

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