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

We introduce a model of one- and two-dimensional (1D and 2D) optical media with the $\chi ^{(2)}$ nonlinearity whose local strength is subject to cusp-shaped spatial modulation, $\chi ^{(2)}\sim r^{-\alpha }$, with $\alpha >0$, 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 $\alpha <1$ and $\alpha <2$, respectively. The 1D solitons have a small instability region, while the 2D solitons have a stability region at $\alpha <0.5$ and are unstable at $\alpha >0.5$. 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.