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A model of the optical media with a spatially structured Kerr nonlinearity isintroduced. The nonlinearity strength is modulated by a set of singular peakson top of a self-focusing or defocusing uniform background. The peaks mayinclude a repulsive or attractive linear potential too. We find that a pair ofmutually symmetric peaks readily gives rise to the spontaneous symmetrybreaking (SSB) of modes pinned to individual peaks, while antisymmetric pinnedmodes are always unstable, transforming into robust spatially odd breathers.Three- and five-peak structures support symmetric modes, with in-phase ortwisted profiles, and do not give rise to asymmetric states. A stability areais found for the twisted states pinned to the triple peaks, while thecorresponding in-phase modes are unstable, unless the three modulation peaksare set very close to each other, covered by a single-peak pinned mode. Allpatterns pinned to five peaks are unstable too. Collisions of moving solitonswith the singular-modulation peak are studied too. Slowly moving solitonsbounce back from the peak, while the collisions are quasi-elastic for fastsolitons. In the intermediate case, the soliton is destroyed by the collision.In a special case, the condition of a resonance of the incident soliton with atrapped mode supported by the peak leads to capture of the soliton.

Solitons supported by singular modulation of the cubic nonlinearity