Spatial control of the competition between self-focusing and self-defocusing nonlinearities in one- and two-dimensional systems

We introduce a system with competing self-focusing (SF) and self-defocusing (SDF) terms, which have the same scaling dimension. In the one-dimensional (1D) system, this setting is provided by a combination of the SF cubic term multiplied by the delta function δ(x) and a spatially uniform SDF quintic term. This system gives rise to the most general family of 1D Townes solitons, with the entire family being unstable. However, it is completely stabilized by a finite-width regularization of the δ function. The results are produced by means of numerical and analytical methods. We also consider the system with a symmetric pair of regularized δ functions, which gives rise to a wealth of symmetric, antisymmetric, and asymmetric solitons, linked by a bifurcation loop, that accounts for the breaking and restoration of the symmetry. Soliton families in two-dimensional (2D) versions of both the single- and double-δ-functional systems are also studied. The 1D and 2D settings may be realized for spatial solitons in optics and in Bose-Einstein condensates.