Some results for a system of NLS arising in optical materials with χ3
nonlinear response
Abstract
In this paper, we investigate the nonlinear Schödinger equations with
cubic interactions, arising in nonlinear optics. To begin, we prove the
existence results for normalized ground state solutions in the L 2
-subcritical case and L 2 -supercritical case respectively. Our proofs
relies on the Concentration-compactness principle, Pohozaev manifold and
rearrangement technique. Then, we establish the nonexistence of
normalized ground state solutions in the L 2 -critical case by finding
that there exists a threshold. In addition, based on the existence of
the normalized solutions, we also establish the blow-up results are
shown by using localized virial estimates, and a new blow-up criterion
which is related to normalized solutions.