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

Stability of soliton families in one-dimensional nonlinear Schrodinger 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. (C) 2016 Elsevier B.V. All rights reserved.