Nonlinear light behaviors near phase transition in non-parity-time-symmetric complex waveguides

Many classes of non-parity-time ($\mathcal{PT}$)-symmetric waveguides with arbitrary gain and loss distributions still possess all-real linear spectrum or exhibit phase transition. In this Letter, nonlinear light behaviors in these complex waveguides are probed analytically near a phase transition. Using multi-scale perturbation methods, a nonlinear ordinary differential equation (ODE) is derived for the light’s amplitude evolution. This ODE predicts that a single class of these non-$\mathcal{PT}$-symmetric waveguides supports soliton families and amplitude-oscillating solutions both above and below linear phase transition, in close analogy with $\mathcal{PT}$-symmetric systems. For the other classes of waveguides, the light’s intensity always amplifies under the effect of nonlinearity, even if the waveguide is below the linear phase transition. These analytical predictions are confirmed by direct computations of the full system.