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arXiv · 1904.09066

Scattering for the $L^2$ supercritical point NLS

Riccardo Adami·Reika Fukuizumi·Justin Holmer
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Abstract

We consider the 1D nonlinear Schr\"odinger equation with focusing point nonlinearity. "Point" means that the pure-power nonlinearity has an inhomogeneous potential and the potential is the delta function supported at the origin. This equation is used to model a Kerr-type medium with a narrow strip in the optic fibre. There are several mathematical studies on this equation and the local/global existence of solution, blow-up occurrence and blow-up profile have been investigated. In this paper we focus on the asymptotic behavior of the global solution, i.e, we show that the global solution scatters as t tends to minus/plus infinity in the $L^2$ supercritical case. The main argument we use is due to Kenig-Merle, but it is required to make use of an appropriate function space (not Strichartz space) according to the smoothing properties of the associated integral equation.

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Riccardo Adami, Reika Fukuizumi, Justin Holmer. 2019-04-19. Scattering for the $L^2$ supercritical point NLS. https://arxiv.org/abs/1904.09066

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