One-dimensional solitons in fractional Schr\"{o}dinger equation with a spatially modulated nonlinearity: nonlinear lattice

TL;DR

This paper demonstrates the existence and stability of one-dimensional bright solitons in a fractional Schrödinger equation with a spatially modulated nonlinear lattice, highlighting the role of the Levy index in soliton stability.

Contribution

It introduces a model with fractional diffraction and nonlinear lattice, analyzing the stability of fundamental and multihump solitons, and identifies the Levy index threshold for stabilization.

Findings

Stable bright solitons exist in the model.

Soliton profiles are affected by the Levy index.

Stability depends on the Levy index exceeding a threshold.

Abstract

The existence and stability of stable bright solitons in one-dimensional (1D) media with a spatially periodical modulated Kerr nonlinearity are demonstrated by means of the linear-stability analysis and in direct numerical simulations. The nonlinear potential landscape can balance the fractional-order diffraction and thus stabilizes the solitons, making the model unique and governed by the recently introduced fractional Schr\"{o}dinger equation with a self-focusing cubic nonlinear lattice. Both 1D fundamental and multihump solitons (in forms of dipole and tripole ones) are found, which occupy one or three cells of the nonlinear lattice respectively, depending on the soliton's power (intensity). We find that the profiles of the predicted soliton families are impacted intensely by the L\'{e}vy index $\alpha$ which denotes the level of fractional Laplacian, so does to their stability. The stabilization of soliton families is possible if $\alpha$ exceeds a threshold value, below which the balance between fractional-order diffraction and the spatially modulated focusing nonlinearity will be broken.