Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sharp conditions for scattering and blow-up for a system of NLS arising
  in optical materials with $\chi^3$ nonlinear response   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.

9
The categories of this paper are: nlin
21
The categories of this paper are: adap-org
22
The categories of this paper are: chao-dyn
6
The categories of this paper are: astro-ph
27
The categories of this paper are: comp-gas
7
The categories of this paper are: hep-th
12
The categories of this paper are: nucl-th
1
The categories of this paper are: cs
19
The categories of this paper are: eess
36
The categories of this paper are: cmp-lg
34
The categories of this paper are: chem-ph
33
The categories of this paper are: plasm-ph
11
The categories of this paper are: hep-lat
35
The categories of this paper are: mtrl-th
24
The categories of this paper are: dg-ga
14
The categories of this paper are: math-ph