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We study a system of inhomogeneous nonlinear Schr\"odinger equations arisingin optical media with a $\chi^{(2)}$ nonlinearity whose local strength issubject to cusp-shaped spatial modulation, $\chi^{(2)}\sim |x|^{-\alpha}$ with$\alpha > 0$, which can be induced by spatially nonuniform poling. We firstestablish a vectorial Gagliardo--Nirenberg type inequality related to thesystem, by which we determine sufficient conditions on initial data leading tothe existence of globally in time solutions of the system. We also providecriteria for the existence of non-radial blow-up solutions with mass-criticaland mass-supercritical nonlinearities. By exploiting the spatial decay of thenonlinearity at infinity, we establish the non-radial energy scattering in themass-supercritical regime for global solutions with initial data lying below amass-energy threshold. We prove, by variational analysis, the existence andqualitative properties of ground states related to the system. The limitingbehavior of positive radial ground states once $\alpha$ approaches zero isstudied by means of the mountain-pass energy. The stability and instability ofground state standing waves are also investigated.

On a system of inhomogeneous nonlinear Schr\"odinger equations arising in optical media with a $\chi^{(2)}$ nonlinearity