Cubic-quintic NLS: scattering beyond the virial threshold

TL;DR

This paper extends the understanding of scattering for the 3D nonlinear Schrödinger equation with combined focusing cubic and defocusing quintic nonlinearities, proving scattering in a broader parameter region beyond previous virial-based results.

Contribution

It establishes scattering results in a larger parameter space where the virial quantity is not necessarily positive, advancing the theory of nonlinear Schrödinger equations.

Findings

Proved scattering in a larger region of the mass/energy plane.

Extended previous results beyond the virial positivity region.

Enhanced understanding of the nonlinear Schrödinger equation dynamics.

Abstract

We consider the nonlinear Schr\"odinger equation in three space dimensions with combined focusing cubic and defocusing quintic nonlinearity. This problem was considered previously by Killip, Oh, Pocovnicu, and Visan, who proved scattering for the whole region of the mass/energy plane where the virial quantity is guaranteed to be positive. In this paper we prove scattering in a larger region where the virial quantity is no longer guaranteed to be sign definite.