Inhomogeneous coupled non-linear Schrödinger systems

This work studies an inhomogeneous Schrödinger coupled system in the mass-super-critical and energy-sub-critical regimes. In the focusing sign, a sharp dichotomy of global existence and scattering vs finite time blow-up of solutions is obtained using some variational methods, a sharp Gagliardo–Nirenberg-type inequality, and a new approach of Dodson and Murphy Proc. Am. Math. Soc. 145(11), 4859–4867 (2017). In the defocusing sign, using a classical Morawetz estimate, the scattering of global solutions in the energy space is proved.