Solitons supported by singular spatial modulation of the Kerr nonlinearity

We introduce a setting based on the one-dimensional nonlinear Schrödinger equation (NLSE) with the self-focusing cubic term modulated by a singular function of the coordinate |x|-α. It may be additionally combined with the uniform self-defocusing (SDF) nonlinear background, and with a similar singular repulsive linear potential. The setting, which can be implemented in optics and Bose-Einstein condensates, aims to extend the general analysis of the existence and stability of solitons in NLSEs. Results for fundamental solitons are obtained analytically and verified numerically. The solitons feature a quasicuspon shape, with the second derivative diverging at the center, and are stable in the entire existence range, which is 0≤α<1. Dipole (odd) solitons are also found. They are unstable in the infinite domain, but stable in the semi-infinite one. In the presence of the SDF background, there are two subfamilies of fundamental solitons, one stable and one unstable, which exist together above a threshold value of the norm (total power of the soliton). The system, which additionally includes the singular repulsive linear potential, emulates solitons in a uniform space of the fractional dimension, 0<D≤1. A two-dimensional extension of the system, based on the quadratic (χ(2)) nonlinearity, is also formulated.