Sharp conditions for scattering and blow-up for a system of NLS arising in optical materials with χ³ nonlinear response

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 virialtype 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], [18], [26].…”

Blow-up results for systems of nonlinear Schrödinger equations with quadratic interaction

“…This is no more the case in the non-mass-resonance setting, i.e. when κ = 1 2 . We refer the reader to [25,Introduction] for an exhaustive list of references in which the effects of the mass and non-mass resonance conditions on the dynamics of solutions to systems similar to (1.4) were studied.…”

“…The proof is based on the concentration/compactness and rigidity scheme in the spirit of Kenig and Merle [20]. Wang and Yang [28] extended the result of [13] to the non-radial case provided that κ belongs to a small neighbourhood of 1 2 . Their proof made use of a recent method of Dodson and Murphy [11] using the interaction Morawetz inequality.…”

“…We refer the reader to the early work of Martel [23] in the context of the NLS equation in anisotropic spaces, and the more recent papers [3,16,17]. See also our recent paper [1] in the context of NLS system with cubic interaction.…”

“…From a pure mathematical perspective, distinguishing the cases κ = 1 2 and 0 < κ = 1 2 plays a role in the virial identities related to (1.4). Under the mass-resonance condition, namely κ = 1 2 , some terms in the virial identities disappear, and the study of the dynamics of solutions is easier due to these cancellations. This is no more the case in the non-mass-resonance setting, i.e.…”