Scattering below the ground state for the 2$d$ radial nonlinear Schrödinger equation

Abstract

We revisit the problem of scattering below the ground state threshold for the mass-supercritical focusing nonlinear Schrödinger equation in two space dimensions. We present a simple new proof that treats the case of radial initial data. The key ingredient is a localized virial/Morawetz estimate; the radial assumption aids in controlling the error terms resulting from the spatial localization. 2D FOCUSING NLS 11

Global dynamics for a class of inhomogeneous nonlinear Schrödinger equations with potential

“…In the case , the energy scattering for the focusing problem (1.6) was first established with and by Farah–Guzmán [18] via the concentration‐compactness argument. Recently, Xu–Zhao [38] and Dinh [9] simultaneously proved the energy scattering for the focusing problem (1.6) with and by adapting a new argument of Arora–Dodson–Murphy [1].…”

Scattering theory for NLS with inverse-square potential in 2D

“…In these works the authors proved via the concentration compactness. Recently, Arora-Dodson-Murphy [2] give a simple proof with radial initial data, which avoids concentration compactness.…”

Long time dynamics for the focusing nonlinear Schrödinger equation with exponential nonlinearities

“…The proof of energy scattering in Theorem 1.2 is based on the Morawetz-Sobolev approach of Dodson and Murphy [22] (see also [2,3,20]). Due to the spatial growth of nonlinearity, we make an intensive use of radial Sobolev embeddings.…”