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 (X⁽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 X⁽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 ≤ N^m (q), all trappedmodes 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 semivortex with topological charge m/2 = 1 /2), at N ≤ N_cm(q), and unstable above this threshold. On the other hand, N_cm(q) = 0 at q = 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 semivortex 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.