Sharp conditions for scattering and blow-up for a system of NLS arising in optical materials with $χ^3$ nonlinear response

Abstract

We study the asymptotic dynamics for solutions to a system of nonlinear Schrödinger equations with cubic interactions, arising in nonlinear optics. We provide sharp threshold criteria leading to global well-posedness and scattering of solutions, as well as formation of singularities in finite time for (anisotropic) symmetric initial data. The free asymptotic results are proved by means of Morawetz and interaction Morawetz estimates. The blow-up results are shown by combining variational analysis and an ODE argument, which overcomes the unavailability of the convexity argument based on virial-type identities.

Scattering for quadratic-type Schrödinger systems in dimension five without mass-resonance

“…Instead, their analysis is based on an interaction Morawetz inequality. In the same direction, scattering for some Schrödinger-type system have appeared, for instance, in [1], [16], [24].…”