Bright solitons from the nonpolynomial Schr\"odinger equation with inhomogeneous defocusing nonlinearities

TL;DR

This paper investigates bright solitons in a 1D nonpolynomial Schrödinger equation with inhomogeneous defocusing nonlinearities, providing analytical and numerical solutions, and analyzing their stability for various modes.

Contribution

It introduces new analytical and numerical solutions for bright solitons in a nonpolynomial SDF model with spatially varying nonlinearity, including stability analysis for different modes.

Findings

Ground states and single-node modes are stable.

Higher-order modes are unstable in the algebraic modulation case.

Unstable states evolve into stable lower-order modes.

Abstract

Extending the recent work on models with spatially nonuniform nonlinearities, we study bright solitons generated by the nonpolynomial self-defocusing (SDF) nonlinearity in the framework of the one-dimensional (1D) Mu\~{n}oz-Mateo - Delgado (MM-D) equation (the 1D reduction of the Gross-Pitaevskii equation with the SDF nonlinearity), with the local strength of the nonlinearity growing at any rate faster than |x| at large values of coordinate x. We produce numerical solutions and analytical ones, obtained by means of the Thomas-Fermi approximation (TFA), for nodeless ground states, and for excited modes with 1, 2, 3, and 4 nodes, in two versions of the model, with the steep (exponential) and mild (algebraic) nonlinear-modulation profiles. In both cases, the ground states and the single-node ones are completely stable, while the stability of the higher-order modes depends on their norm (in the case of the algebraic modulation, they are fully unstable). Unstable states spontaneously evolve into their stable lower-order counterparts.