Stability of localized modes in PT-symmetric nonlinear potentials

TL;DR

This paper investigates the stability of localized modes in nonlinear Schrödinger equations with PT-symmetric potentials, revealing how potential shape and mode width influence stability through Evans function analysis and numerical simulations.

Contribution

It provides a detailed stability analysis of localized modes in PT-symmetric nonlinear potentials, highlighting the role of potential shape and mode width in stability criteria.

Findings

Localized modes become stable above a certain propagation constant threshold.

Mode stability depends on the relation between mode width and potential size.

Numerical simulations confirm Evans function predictions.

Abstract

We report on detailed investigation of the stability of localized modes in the nonlinear Schrodinger equations with a nonlinear parity-time (alias PT) symmetric potential. We are particularly focusing on the case where the spatially-dependent nonlinearity is purely imaginary. We compute the Evans function of the linear operator determining the linear stability of localized modes. Results of the Evans function analysis predict that for sufficiently small dissipation localized modes become stable when the propagation constant exceeds certain threshold value. This is the case for periodic and $\tanh$-shaped complex potentials where the modes having widths comparable with or smaller than the characteristic width of the complex potential are stable, while broad modes are unstable. In contrast, in complex potentials that change linearly with transverse coordinate all modes are stable, what suggests that the relation between width of the modes and spatial size of the complex potential define the stability in the general case. These results were confirmed using the direct propagation of the solutions for the mentioned examples.