# Attraction centers and PT-symmetric delta-functional dipoles in critical and supercritical self-focusing media

## TL;DR

This paper introduces a model of solitons in critical and supercritical self-focusing media with PT-symmetric defects, demonstrating stabilization and stability conditions, and explores their dynamics and stability through analytical and numerical methods.

## Contribution

The study provides exact solutions for pinned solitons with delta-function defects in a NLSE with critical and supercritical nonlinearities, including stability analysis under PT-symmetric gain and loss.

## Key findings

- Pinned solitons are stabilized by attractive defects.
- Entire family of solitons is stable in the quintic medium without gain-loss.
- Unstable solitons can transform into breathers or collapse.

## Abstract

We introduce a model based on the one-dimensional nonlinear Schroedinger equation (NLSE) with the critical (quintic) or supercritical self-focusing nonlinearity. We demonstrate that a family of solitons, which are unstable in this setting against collapse, is stabilized by pinning to an attractive defect, that may also include a parity-time (PT)-symmetric gain-loss component. The model can be realized as a planar waveguide in optics, and in a super-Tonks-Girardeau bosonic gas. For the attractive defect with the delta-functional profile, a full family of the pinned solitons is found in an exact form. In the absence of the gain-loss term, the solitons' stability is investigated analytically too, by means of the Vakhitov-Kolokolov criterion; in the presence of the PT-balanced gain and loss, the stability is explored by means of numerical methods. In particular, the entire family of pinned solitons is stable in the quintic medium if the gain-loss term is absent. A stability region for the pinned solitons persists in the model with an arbitrarily high power of the self-focusing nonlinearity. A weak gain-loss component gives rise to alternations of stability and instability in the system's parameter plane. Those solitons which are unstable under the action of the supercritical self-attraction are destroyed by the collapse. If the self-attraction-driven instability is weak and the gain-loss term is present, unstable solitons spontaneously transform into localized breathers. The same outcome may be caused by a combination of the critical nonlinearity with the gain and loss. Instability of the solitons is also possible when the PT-symmetric gain-loss term is added to the subcritical nonlinearity. The system with self-repulsive nonlinearity is briefly considered too, producing completely stable families of pinned localized states.