Going Beyond the Threshold: Scattering and Blow-up in the Focusing NLS Equation

We study the focusing nonlinear Schrodinger equation i partial derivative(t)u+Delta u+vertical bar u vertical bar(p-1)u = 0, x is an element of R-N 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 the mass-energy threshold for finite variance solutions in the energy-subcritical and energy-critical regimes, obtaining scattering and blow-up criteria for solutions with arbitrary 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.