A system of inhomogeneous NLS arising in optical media with a $\chi^{(2)}$ nonlinearity, part I : Dynamics

Abstract

We study a system of inhomogeneous nonlinear Schr\"odinger equations that emerge in optical media with a $\chi^{(2)}$ nonlinearity. This nonlinearity, whose local strength is subject to a cusp-shaped spatial modulation, $\chi^{(2)}\sim |x|^{-\alpha}$ where $\alpha > 0$, can be induced by spatially non-uniform poling. Our first step is to establish a vectorial Gagliardo--Nirenberg type inequality related to the system. This allows us to identify the necessary conditions on the initial data that lead to the existence of global in time solutions. By exploiting the spatial decay at infinity of the nonlinearity, we demonstrate the non-radial energy scattering in the mass-supercritical regime for global solutions. These solutions have initial data that lie below a mass-energy threshold, regardless of whether the system is mass-resonant or non-mass resonant. Lastly, we provide the criteria for the existence of non-radial blow-up solutions with mass-critical and mass-supercritical nonlinearities in both mass and non-mass resonance cases.