Scattering for the mass super-critical perturbations of the mass critical nonlinear Schr\"odinger equations

TL;DR

This paper proves global well-posedness and scattering for a class of nonlinear Schrödinger equations with mixed mass-critical and supercritical nonlinearities, including the 2D cubic-quintic case, for non-radial data in dimensions 1 to 4.

Contribution

It establishes the first scattering results for these equations with non-radial data below the energy threshold, extending previous radial symmetry results.

Findings

Global well-posedness for the considered equations.

Scattering results for non-radial data in dimensions 1 to 4.

Includes the two-dimensional cubic-quintic nonlinear Schrödinger equation.

Abstract

We consider the Cauchy problem for the nonlinear Schr\"odinger equation with double nonlinearities with opposite sign, with one term is mass-critical and the other term is mass-supercritical and energy-subcritical, which includes the famous two-dimensional cubic-quintic nonlinear Schr\"odinger equaton. We prove global wellposedness and scattering in $H^1(\mathbb{R}^d)$ below the threshold for non-radial data when $1 \le d \le 4$.