Two-dimensional solitons in second-harmonic-generating media with fractional diffraction

We introduce a system of propagation equations for the fundamental-frequency (FF) and second-harmonic (SH) waves in the bulk waveguide with the effective fractional diffraction and quadratic (χ⁽²⁾) nonlinearity. The numerical solution produces families of ground-state (zero-vorticity) two-dimensional solitons in the free space, which are stable in exact agreement with the Vakhitov–Kolokolov criterion, while vortex solitons are completely unstable in that case. Mobility of the stable solitons and inelastic collisions between them are briefly considered too. In the presence of a harmonic-oscillator (HO) trapping potential, families of partially stable single- and two-color solitons (SH-only or FF-SH ones, respectively) are obtained, with zero and nonzero vorticities. The single- and two-color solitons are linked by a bifurcation which takes place with the increase of the soliton's power.