Stabilizing single- and two-color vortex beams in quadratic media by a trapping potential

We consider two-dimensional (2D) localized modes in the second-harmonic-generating \chi ^{(2)} system with the harmonic-oscillator (HO) trapping potential. In addition to its realization in optics, the system describes the mean-field dynamics of mixed atomic-molecular Bose-Einstein condensates (BECs). The existence and stability of various modes is determined by their total power, N, topological charge, m/2 [m is the intrinsic vorticity of the second-harmonic (SH) field], and \chi ^{(2)} mismatch, q. The analysis is carried out in a numerical form and, in parallel, by means of the variational approximation (VA), which produces results that agree well with numerical findings. Below a certain power threshold, N\leq N_{c}^{(m)}(q), all trapped modes are of the single-color type, represented by the SH component only, while the fundamental-frequency (FF) one is absent. In contrast with the usual situation, where such modes are always unstable, we demonstrate\ that they are stable, for m=0,1,2 (the mode with m=1 may be formally considered as a semi-vortex with topological charge m/2=1/2), at N\leq N_{c}^{(m)}(q), and unstable above this threshold. On the other hand, N_{c}^{(m)}(q)\equiv 0 at q\geq q_{\max} (in our notation, q_{\max}=1$), hence the single-color modes are unstable in the latter case. At N=N_{c}^{(m)}, the modes with m=0 and m=2 undergo a pitchfork bifurcation, which gives rise to two-color states, which remain completely stable for m=0. The two-color vortices with m=2 (topological charge 1) have an upper stability border, N=N_{c2}(q). Above the border, they exhibit periodic splittings and recombinations, while keeping their vorticity. The semi-vortex does not bifurcate; at N=N_{c}^{(m=1)}, it exhibits quasi-chaotic oscillations and a rotating "groove" resembling a screw-edge dislocation induced by the semi-integer vorticity.