Small-amplitude nonlinear modes under the combined effect of the parabolic potential, nonlocality and ${\cal PT}$ symmetry

TL;DR

This paper investigates small-amplitude nonlinear modes in a nonlocal nonlinear Schrödinger equation with PT-symmetric parabolic potential, revealing how PT symmetry and nonlocality influence their stability.

Contribution

It demonstrates the existence of continuous families of nonlinear modes and analyzes how PT symmetry and nonlocality affect their stability, including the impact of the nonlocal kernel shape.

Findings

PT symmetry or nonlocality can stabilize small-amplitude modes

Stability depends on the shape of the nonlocal kernel

Results extend to finite-amplitude modes

Abstract

We consider nonlinear modes of the nonlinear Schr\"odinger equation with a nonlocal nonlinearity and an additional PT-symmetric parabolic potential. We show that there exists a set of continuous families of nonlinear modes and study their linear stability in the limit of small nonlinearity. It is demonstrated that either PT symmetry or the nonlocality can be used to manage the stability of the small-amplitude nonlinear modes. The stability properties are also found to depend on the particular shape of the nonlocal kernel. Additional numerical simulations show that the stability results remain valid not only for the infinitesimally small nonlinear modes, but also for the modes of finite amplitude