Going beyond the threshold: scattering and blow-up in the focusing NLS equation

TL;DR

This paper analyzes the focusing nonlinear Schrödinger equation in the supercritical regime, extending the understanding of scattering and blow-up phenomena for solutions with large energy and mass, including criteria applicable beyond the critical cases.

Contribution

It extends the scattering versus blow-up dichotomy above the energy threshold and provides new blow-up criteria applicable in the supercritical regime.

Findings

Extended scattering and blow-up criteria for large energy solutions

Characterized behavior of ground state data with quadratic phase

Provided examples illustrating the theoretical results

Abstract

We study the focusing nonlinear Schr\"odinger equation in the $L^2$-supercritical regime with finite energy and finite variance initial data. We investigate solutions above the energy (or mass-energy) threshold. In our first result, we extend the known scattering versus blow-up dichotomy above that threshold for finite variance solutions in the energy-subcritical and energy-critical regimes, obtaining scattering and blow-up criteria for solutions with arbitrarily large mass and energy. As a consequence, we characterize the behavior of the ground state initial data modulated by a quadratic phase. Our second result gives two blow up criteria, which are also applicable in the energy-supercritical NLS setting. We finish with various examples illustrating our results.