Stabilization of three-wave vortex beams in the waveguide

We consider two-dimensional (2D) localized vortical modes in the three-wave system with the quadratic ($\chi ^{(2)}$) nonlinearity, alias nondegenerate second-harmonic-generating system, guided by the isotropic harmonic-oscillator (HO) (alias parabolic) confining potential. In addition to the straightforward realization in optics, the system models mixed atomic-molecular Bose-Einstein condensates (BECs). The main issue is stability of the vortex modes, which is investigated through computation of instability growth rates for eigenmodes of small perturbations, and by means of direct simulations. The threshold of parametric instability for single-color beams, represented solely by the second harmonic (SH) with zero vorticity, is found in an analytical form with the help of the variational approximation (VA). Trapped states with vorticities $\left( +1,-1,0\right)$ in the two fundamental-frequency (FF) components and the SH one [the so-called \textit{hidden-vorticity} (HV) modes] are completely unstable. Also unstable are \textit{semi-vortices} (SVs), with component vorticities $$. However, full vortices, with charges $\left( 1,1,2\right)$, have a well-defined stability region. Unstable full vortices feature regions of robust dynamical behavior, where they periodically split and recombine, keeping their vortical content.