On a system of inhomogeneous nonlinear Schrödinger equations arising in optical media with a X(2) nonlinearity

We study a system of inhomogeneous nonlinear Schrödinger equations arising in optical media with a χ (2) nonlinearity whose local strength is subject to cusp-shaped spatial modulation, χ (2) ∼ |x| −α with α > 0, which can be induced by spatially nonuniform poling. We first establish a vectorial Gagliardo-Nirenberg type inequality related to the system, by which we determine sufficient conditions on initial data leading to the existence of globally in time solutions of the system. We also provide criteria for the existence of non-radial blow-up solutions with mass-critical and mass-supercritical nonlinearities. By exploiting the spatial decay of the nonlinearity at infinity, we establish the non-radial energy scattering in the mass-supercritical regime for global solutions with initial data lying below a mass-energy threshold. We prove, by variational analysis, the existence and qualitative properties of ground states related to the system. The limiting behavior of positive radial ground states once α approaches zero is studied by means of the mountain-pass energy. The stability and instability of ground state standing waves are also investigated. Content