Stability Analysis of Fixed points in a Parity-time symmetric coupler with Kerr nonlinearity

We report our study on nonlinear parity-time (PT) symmetric coupler from a dynamical perspective. In the linear regime, the differential equations governing the dynamics of the coupler, under some parametric changes, can be solved exactly. But with the inclusion of nonlinearity, analytical solution of the system is a rather complicated job. And the sensitiveness of the system on the initial conditions is yet another critical issue. To circumvent the situation, we have employed the mathematical framework of nonlinear dynamics. Considering the parity-time threshold of the linear PT-coupler as the reference point, we find that in nonlinear coupler the parity-time symmetric threshold governs the existence of fixed points. We have found that the stability of the ground state undergoes a phase transition when the gain/loss coefficient is increased from zero to beyond the PT threshold. In the unbroken PT regime, we find that the instabilities in the initial launch conditions can trigger an exponential growth and decay of optical power in the waveguides. The attractor and chaotic behavior of the system has also been studied. We ascertain that the coupler exhibits selfstabilizing characteristics. It is shown that the chaotic behavior could be controlled with judicious choice of waveguide parameters.