Scattering for the 𝐿² supercritical point NLS

Adami, Riccardo; Fukuizumi, Reika; Holmer, Justin

doi:10.1090/tran/8065

Scattering for the $L^2$ supercritical point NLS
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by Riccardo Adami, Reika Fukuizumi and Justin Holmer PDF

Trans. Amer. Math. Soc. 374 (2021), 35-60 Request permission

Abstract:

We consider the 1D nonlinear Schrödinger 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 a 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 $t\to \pm \infty$ 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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