Ali Mostafazadeh

We derive certain identities satisfied by the left/right-reflection and transmission amplitudes, \(R^{l/r}(k)\) and \(T(k)\), of general \({\cal PT}\)-symmetric scattering potentials. We use these identities to give a general proof of the relations, \(|T(-k)|=|T(k)|\) and \(|R^r(-k)|=|R^l(k)|\), conjectured in [Z. Ahmed, J. Phys. A 45 (2012) 032004], establish the generalized unitarity relation: \(R^{l/r}(k)R^{l/r}(-k)+|T(k)|^2=1\), and show that it is a common property of both real and complex \({\cal PT}\)-symmetric potentials. The same holds for \(T(-k)=T(k)^*\) and \(|R^r(-k)|=|R^l(k)|\).

http://arxiv.org/abs/1405.4212

Quantum Physics (quant-ph); Mathematical Physics (math-ph); Optics (physics.optics)

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)

Amarendra K. Sarma

We carry out a modulation instability (MI) analysis in nonlinear complex parity-time (PT) symmetric periodic structures. All the three regimes defined by the PT-symmetry breaking point or threshold, namely, below threshold, at threshold and above threshold are discussed. It is found that MI exists even beyond the PT-symmetry threshold indicating the possible existence of solitons or solitary waves, in conformity with some recent reports. We find that MI does not exist at the PT-symmetry breaking point in the case of normal dispersion below a certain nonlinear threshold. However, in the case of anomalous dispersion regime, MI does exist even at the PT-symmetry breaking point.

http://arxiv.org/abs/1405.2706

Optics (physics.optics)