Localized modes in PT-symmetric sine-Gordon couplers with phase shift

In this article, we address the nonlinear localized modes for waveguide arrays with parity-time (PT)-symmetry characterized by the balanced gain and loss parameters. To investigate the PT-symmetric nonlinear modes, the system of coupled sine-Gordon equations together with a variety of nonlinear terms are considered. We applied asymptotic analysis together with multiple-scale expansion and obtained analytical amplitude equations for the optical modes, including the fundamental and excited modes. The nonlinear localized modes are obtained by considering the discrete and spatially continuous spectrum. It is noted that the obtained amplitude equations of oscillations are affected by the strength of cross-coupling, that is, when the system is coupled weakly, the amplitude in the lossy side decreases continuously while it increases on the gain side of the waveguide for a long time. It is further noted that in the presence of strong and sinusoidally coupling when the damping is present, the rate of blow and decay re-balance the waveguides, and the system oscillates for a long period. It is observed that, in the case of quadratic coupling, the amplitude in the lossy side decreases while the attenuated field increases. It is also observed that when the waveguide is coupled strongly, the interaction in kink and anti-kink oscillations form a breather. The analytical results are compared with the obtained numerical simulations in the numerical section, where good agreements are obtained.