One- and two-dimensional solitons supported by singular modulation of quadratic nonlinearity

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We introduce a model of one- and two-dimensional (1D and 2D) optical media with the χ⁽²⁾ nonlinearity whose local strength is subject to cusp-shaped spatial modulation, χ⁽²⁾ r⁻α, with α>0, which can be induced by spatially nonuniform poling. Using analytical and numerical methods, we demonstrate that this setting supports 1D and 2D fundamental solitons, at α<1 and α<2, respectively. The 1D solitons have a small instability region, while the 2D solitons have a stability region at α<0.5 and are unstable at α>0.5. 2D solitary vortices are found too. They are unstable, splitting into a set of fragments, which eventually merge into a single fundamental soliton pinned to the cusp. Spontaneous symmetry breaking of solitons is studied in the 1D system with a symmetric pair of the cusp-modulation peaks.