Ring-Shaped Gap Solitons and Vortices in Saturable Nonlinear Media

We demonstrate that ring-shaped gap solitons can be stabilized in the nonlinear Schrödinger equation with saturable nonlinearity and a radially periodic linear lattice, encompassing both fundamental solitons and their vortex counterparts. Two types of solitons, distinguished by different numbers of ring peaks—specifically, single-ring and three-ring solitons, as well as their vortex analogues—are studied. By solving the governing equation, we establish the existence, properties, and stability of these localized solutions, complemented with direct propagation simulations. The power of all these soliton families decreases as the propagation constant increases, satisfying the anti-Vakhitov–Kolokolov criterion for soliton stability. Both stable and unstable examples of soliton propagation are displayed. Although they resemble truncated linear Bloch waves, the profiles of these annular solitons are intriguing, as minor peak rings become evident near the edges of the band gaps, causing instabilities. Notably, the phase profiles of ring-shaped vortex solitons differ from the conventional ones; that is, they exhibit spiraling patterns in the central region that alternate periodically in the radial direction.