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

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 $(\lambda ,-\lambda ,{\lambda }^{\ast },-{\lambda }^{\ast })$, 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.

9 pages, 6 figures

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• Waveguides with Absorbing Boundaries: Nonlinearity Controlled by an Exceptional Point and Solitons
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• Effects of the third-order dispersion on continuous waves in complex potentials
• Response of exact solutions of the nonlinear Schrodinger equation to small perturbations in a class of complex external potentials having supersymmetry and parity-time symmetry
• Stability restoration by asymmetric nonlinear states in non-Hermitian double-well potentials
• Nonlinear Schrödinger equations with amplitude-dependent Wadati potentials
• A systematic construction of parity-time ($\mathcal{P}\mathcal{T}$)-symmetric and non-$\mathcal{P}\mathcal{T}$-symmetric complex potentials from the solutions of various real nonlinear evolution equations
• On non-existence of continuous families of stationary nonlinear modes for a class of complex potentials
• Analytical construction of soliton families in one- and two-dimensional nonlinear Schrödinger equations with non-parity-time-symmetric complex potentials