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

TL;DR

This paper establishes precise conditions under which solutions to a system of nonlinear Schrödinger equations in optical materials either scatter or blow up, using advanced analytical techniques.

Contribution

It provides sharp threshold criteria for global existence, scattering, and blow-up in a system of NLS with cubic interactions relevant to nonlinear optics.

Findings

Sharp threshold criteria for scattering and blow-up.

Use of Morawetz estimates for asymptotic analysis.

Variational and ODE methods for blow-up results.

Abstract

We study the asymptotic dynamics for solutions to a system of nonlinear Schr\"odinger 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.