Stabilization of solitons under competing nonlinearities by external potentials

TL;DR

This paper investigates the stability of one-dimensional trapped solitons influenced by competing nonlinearities and external potentials, using analytical and numerical methods applicable to optical and matter-wave systems.

Contribution

It provides a comprehensive analysis of soliton stability under competing nonlinearities with harmonic and delta potentials, including new analytical solutions and stability criteria.

Findings

Stable solitons identified under specific conditions

Analytical solutions for delta-functional potential verified numerically

VK and anti-VK criteria effectively predict stability

Abstract

We report results of the analysis for families of one-dimensional (1D) trapped solitons, created by competing self-focusing (SF) quintic and self-defocusing (SDF) cubic nonlinear terms. Two trapping potentials are considered, the harmonic-oscillator (HO) and delta-functional ones. The models apply to optical solitons in colloidal waveguides and other photonic media, and to matter-wave solitons in Bose-Einstein condensates (BEC) loaded into a quasi-1D trap. For the HO potential, the results are obtained in an approximate form, using the variational and Thomas-Fermi approximations (VA and TFA), and in a full numerical form, including the ground state and the first antisymmetric excited one. For the delta-functional attractive potential, the results are produced in a fully analytical form, and verified by means of numerical methods. Both exponentially localized solitons and weakly localized trapped modes are found for the delta-functional potential. The most essential conclusions concern the applicability of competing Vakhitov-Kolokolov (VK) and anti-VK criteria to the identification of the stability of solitons created under the action of the competing SF and SDF terms.