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

TL;DR

This paper investigates how competing self-focusing and self-defocusing nonlinearities affect soliton stability in one- and two-dimensional systems, revealing stabilization mechanisms and symmetry-breaking phenomena through numerical and analytical methods.

Contribution

It introduces a novel system with competing nonlinearities, analyzes the stabilization of Townes solitons, and explores symmetry-breaking bifurcations in both 1D and 2D settings.

Findings

Regularization stabilizes Townes solitons.

Symmetry-breaking bifurcations occur in double-delta systems.

Soliton families are characterized in 1D and 2D systems.

Abstract

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, $\delta (x)$, and a spatially uniform SDF quintic term. This system gives rise to the most general family of 1D-Townes solitons, the entire family being unstable. However, it is completely stabilized by a finite-width regularization of the $\delta $-function. The results are produced by means of numerical and analytical methods. We also consider the system with a symmetric pair of regularized $\delta $-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 2D versions of both the single- and double-delta-functional systems are also studied. The 1D and 2D settings may be realized for spatial solitons in optics, and in Bose-Einstein condensates.