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

Abstract

We study the focusing nonlinear Schrödinger 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.

To appear in CMP

References in corpus (9)

• Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrodinger equation in the radial case
• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation
• Blowup and Scattering problems for the Nonlinear Schrödinger equations
• Scattering for the non-radial 3D cubic nonlinear Schroedinger equation
• Energy-supercritical NLS: critical ${\dot{H}}^s$-bounds imply scattering
• On blow-up solutions to the 3D cubic nonlinear Schroedinger equation
• Global Behavior Of Finite Energy Solutions To The $d$-Dimensional Focusing Nonlinear Schrödinger Equation
• Scattering for the focusing energy-subcritical NLS
• Numerical simulations of the energy- supercritical Nonlinear Schrödinger equation

Cited by in corpus (5)

• Threshold solutions in the focusing 3D cubic NLS equation outside a strictly convex obstacle
• Exact analytical solution of a novel modified nonlinear Schrödinger equation: solitary quantum waves on a lattice
• On well-posedness and blow-up in the generalized Hartree equation
• Behavior of solutions to the 1D focusing stochastic nonlinear Schrödinger equation with spatially correlated noise
• Blow-up and scattering for the 1D NLS with point nonlinearity above the mass-energy threshold