Stable two-dimensional soliton supported by a local nonlinearity

We show that an attractive nonlinear potential in the form of a modified Azbel two-dimensional (2D) delta function (that, unlike the traditional 2D delta function, gives rise to a well-defined bound state in quantum mechanics) supports a 2D localized state ("soliton"), that may be stable according to the Vakhitov-Kolokolov (VK) criterion, provided that the soliton's amplitude is not too small. A direct, although not general, dynamical stability consideration yields results in compliance with the VK criterion.