We consider a scaling limit of a nonlinear Schr"odinger 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 egin{equation*} ipd{}{t}psi^e(t)= -Delta psi^e(t) + g(arepsilon,mu,|( ho^e,psi^e(t))|^{2mu}) ( ho^e,psi^e(t)) ho^e end{equation*} where $ ho^{e} o delta_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.

The point-like limit for a NLS equation with concentrated nonlinearity in dimension three / Cacciapuoti, Claudio; Finco, Domenico; Noja, Diego; Teta, Alessandro. - In: JOURNAL OF FUNCTIONAL ANALYSIS. - ISSN 0022-1236. - STAMPA. - (2017), pp. ...-.... [10.1016/j.jfa.2017.04.011]

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

TETA, Alessandro
2017

Abstract

We consider a scaling limit of a nonlinear Schr"odinger 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 egin{equation*} ipd{}{t}psi^e(t)= -Delta psi^e(t) + g(arepsilon,mu,|( ho^e,psi^e(t))|^{2mu}) ( ho^e,psi^e(t)) ho^e end{equation*} where $ ho^{e} o delta_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.
2017
nonlinear Schroedinger equation; nonlinear delta interactions; scaling limit
01 Pubblicazione su rivista::01a Articolo in rivista
The point-like limit for a NLS equation with concentrated nonlinearity in dimension three / Cacciapuoti, Claudio; Finco, Domenico; Noja, Diego; Teta, Alessandro. - In: JOURNAL OF FUNCTIONAL ANALYSIS. - ISSN 0022-1236. - STAMPA. - (2017), pp. ...-.... [10.1016/j.jfa.2017.04.011]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/961897
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