Solitons supported by singular spatial modulation of the Kerr nonlinearity

TL;DR

This paper investigates the existence and stability of solitons in a 1D nonlinear Schrödinger equation with a singular spatial modulation of the Kerr nonlinearity, revealing stable quasi-cuspon solitons and their properties.

Contribution

It introduces a novel model with singular nonlinear modulation, analyzes fundamental and dipole solitons, and extends the framework to fractional dimensions and 2D systems.

Findings

Fundamental solitons are stable across the entire existence range (0 < a < 1).

Dipole solitons are unstable in infinite domains but stable in semi-infinite domains.

Two subfamilies of solitons exist with different stability properties in the presence of a self-defocusing background.

Abstract

We introduce a setting based on the one-dimensional (1D) nonlinear Schroedinger equation (NLSE) with the self-focusing (SF) cubic term modulated by a singular function of the coordinate, |x|^{-a}. 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 BEC, 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 quasi-cuspon shape, with the second derivative diverging at the center, and are stable in the entire existence range, which is 0 < a < 1. Dipole (odd) solitons are found too. 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 nonlinearity, is formulated too.