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

36 pages. Communications in Partial Differential Equations, to appear

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Cited by in corpus (3)

• On finite time blow-up for a 3D Davey-Stewartson system
• Standing waves for a Schrödinger system with three waves interaction
• On the energy-critical quadratic nonlinear Schrödinger system with three waves