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The propagation dynamics of two-dimensional (2D) ring-Airy beams is studiedin the framework of the fractional Schr\"{o}dinger equation, which includessaturable or cubic self-focusing or defocusing nonlinearity and L\'{e}vy index((LI) alias for the fractionality) taking values $1\leq\alpha \leq 2$. Themodel applies to light propagation in a chain of optical cavities emulatingfractional diffraction. Management is included by making the diffraction and/ornonlinearity coefficients periodic functions of the propagation distance,$\zeta$. The management format with the nonlinearity coefficient decaying as$1/\zeta$ is considered, too. These management schemes maintain stablepropagation of the ring-Airy beams, which maintain their axial symmetry, incontrast to the symmetry-breaking splitting instability of ring-shaped patternsin 2D Kerr media. The instability driven by supercritical collapse at allvalues $\alpha < 2$ in the presence of the self-focusing cubic term iseliminated, too, by the means of management.

Stabilization of Axisymmetric Airy Beams by Means of Diffraction and  Nonlinearity Management in Two-Dimensional Fractional Nonlinear  Schr\"{o}dinger Equations