Sean Nixon, Jianke Yang

Light propagation in optical waveguides with periodically modulated index of refraction and alternating gain and loss are investigated for linear and nonlinear systems. Based on a multiscale perturbation analysis, it is shown that for many non-parity-time (PT) symmetric waveguides, their linear spectrum is partially complex, thus light exponentially grows or decays upon propagation, and this growth or delay is not altered by nonlinearity. However, several classes of non-PT-symmetric waveguides are also identified to possess all-real linear spectrum. In the nonlinear regime longitudinally periodic and transversely quasi-localized modes are found for PT-symmetric waveguides both above and below phase transition. These nonlinear modes are stable under evolution and can develop from initially weak initial conditions.

http://arxiv.org/abs/1412.6113

Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)

Jianke Yang

Symmetry breaking of solitons in a class of one-dimensional parity-time (PT) symmetric complex potentials with cubic nonlinearity is reported. In generic PT symmetric potentials, such symmetry breaking is forbidden. However, in a special class of PT-symmetric potentials \(V(x)=g^2(x)+αg(x)+ig′(x)\), where \(g(x)\) is a real and even function and α a real constant, symmetry breaking of solitons can occur. That is, a branch of non-PT-symmetric solitons can bifurcate out from the base branch of PT-symmetric solitons when the base branch’s power reaches a certain threshold. At the bifurcation point, the base branch changes stability, and the bifurcated branch can be stable.

http://arxiv.org/abs/1408.0687

Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)

Sean Nixon, Jianke Yang

Solitons in one-dimensional parity-time (PT)-symmetric periodic potentials are studied using exponential asymptotics. The new feature of this exponential asymptotics is that, unlike conservative periodic potentials, the inner and outer integral equations arising in this analysis are both coupled systems due to complex-valued solitons. Solving these coupled systems, we show that two soliton families bifurcate out from each Bloch-band edge for either self-focusing or self-defocusing nonlinearity. An asymptotic expression for the eigenvalues associated with the linear stability of these soliton families is also derived. This formula shows that one of these two soliton families near band edges is always unstable, while the other can be stable. In addition, infinite families of PT-symmetric multi-soliton bound states are constructed by matching the exponentially small tails from two neighboring solitons. These analytical predictions are compared with numerics. Overall agreements are observed, and minor differences explained.

http://arxiv.org/abs/1405.2827

Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)

Jianke Yang

Multi-dimensional complex optical potentials with partial parity-time (PT) symmetry are proposed. The usual PT symmetry requires that the potential is invariant under complex conjugation and simultaneous reflection in all spatial directions. However, we show that if the potential is only partially PT-symmetric, i.e., it is invariant under complex conjugation and reflection in a single spatial direction, then it can also possess all-real spectra and continuous families of solitons. These results are established analytically and corroborated numerically.

http://arxiv.org/abs/1312.3660

Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)

Jianke Yang

For the one-dimensional nonlinear Schroedinger equation with a complex potential, it is shown that if this potential is not parity-time (PT) symmetric, then no continuous families of solitons can bifurcate out from linear guided modes, even if the linear spectrum of this potential is all real. Both localized and periodic non-PT-symmetric potentials are considered, and the analytical conclusion is corroborated by explicit examples. Based on this result, it is argued that PT-symmetry of a one-dimensional complex potential is a necessary condition for the existence of soliton families.

http://arxiv.org/abs/1310.4490

Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)

Jianke Yang

For the one-dimensional nonlinear Schroedinger equations with parity-time (PT) symmetric potentials, it is shown that when a real symmetric potential is perturbed by weak PT-symmetric perturbations, continuous families of asymmetric solitary waves in the real potential are destroyed. It is also shown that in the same model with a general PT-symmetric potential, symmetry breaking of PT-symmetric solitary waves do not occur. Based on these findings, it is conjectured that one-dimensional PT-symmetric potentials cannot support continuous families of non-PT-symmetric solitary waves.

http://arxiv.org/abs/1309.1652

Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)