Existence of coupled optical vortex solitons propagating in a quadratic nonlinear medium

We consider the coupled propagation of an optical field and its second harmonic in a quadratic nonlinear medium governed by a coupled system of Schrodinger equations. We prove the existence of ring-profiled optical vortex solitons appearing as solutions to a constrained minimization problem and as solutions to a min-max problem. In the case of the constrained minimization problem, solutions are shown to be positive with undetermined wave propagation constants, but in the min-max approach, the wave propagation constants can be prescribed. The quadratic nonlinearity introduces some interesting properties not commonly observed in other coupled systems in the context of nonlinear optics, such as the system not accepting any semi-trivial solutions, meaning that optical solitons cannot be observed when, say, one of the beams is off. Additionally, the second harmonic always remains positive.