Stability of soliton families in nonlinear Schroedinger equations with non-parity-time-symmetric complex potentials

TL;DR

This paper investigates the stability of solitons in nonlinear Schrödinger equations with non-PT-symmetric complex potentials, revealing stable regimes, bifurcations, and surprising eigenvalue symmetries that influence soliton stability.

Contribution

It demonstrates that solitons can be stable in non-PT-symmetric systems and uncovers a novel eigenvalue quartet symmetry affecting their stability.

Findings

Solitons are stable over a wide parameter range.

A pseudo-Hamiltonian-Hopf bifurcation causes oscillatory instability.

Eigenvalues form surprising quartets similar to conservative systems.

Abstract

Stability of soliton families in one-dimensional nonlinear Schroedinger equations with non-parity-time (PT)-symmetric complex potentials is investigated numerically. It is shown that these solitons can be linearly stable in a wide range of parameter values both below and above phase transition. In addition, a pseudo-Hamiltonian-Hopf bifurcation is revealed, where pairs of purely-imaginary eigenvalues in the linear-stability spectra of solitons collide and bifurcate off the imaginary axis, creating oscillatory instability, which resembles Hamiltonian-Hopf bifurcations of solitons in Hamiltonian systems even though the present system is dissipative and non-Hamiltonian. The most important numerical finding is that, eigenvalues of linear-stability operators of these solitons appear in quartets $(\lambda, -\lambda, \lambda^*, -\lambda^*)$, similar to conservative systems and PT-symmetric systems. This quartet eigenvalue symmetry is very surprising for non-PT-symmetric systems, and it has far-reaching consequences on the stability behaviors of solitons.