  We investigate two-dimensional (2D) states of Bose-Einstein condensates (BEC)
with self-attraction or self-repulsion, trapped in an axially symmetric
optical-lattice potential periodic along the radius. Unlike previously studied
2D models with Bessel lattices, no localized states exist in the linear limit
of the present model, hence all localized states are truly nonlinear ones. We
consider the states trapped in the central potential well, and in remote
circular troughs. In both cases, a new species, in the form of \textit{radial
gap solitons}, are found in the repulsive model (the gap soliton trapped in a
circular trough may additionally support stable dark-soliton pairs). In remote
troughs, stable localized states may assume a ring-like shape, or shrink into
strongly localized solitons. The existence of stable annular states, both
azimuthally uniform and weakly modulated ones, is corroborated by simulations
of the corresponding Gross-Pitaevskii equation. Dynamics of strongly localized
solitons circulating in the troughs is also studied. While the solitons with
sufficiently small velocities are stable, fast solitons gradually decay, due to
the leakage of matter into the adjacent trough under the action of the
centrifugal force. Collisions between solitons are investigated too. Head-on
collisions of in-phase solitons lead to the collapse; $\pi $-out of phase
solitons bounce many times, but eventually merge into a single soliton without
collapsing. The proposed setting may also be realized in terms of spatial
solitons in photonic-crystal fibers with a radial structure.

Other Condensed Matter;Pattern Formation and Solitons