Stable and chaotic solutions of the complex Ginzburg-Landau equation with periodic boundary conditions

We study, analytically and numerically, the dynamical behavior of the solutions of the complex Ginzburg-Landau equation with diffraction but without diffusion, which governs the spatial evolution of the field in an active nonlinear laser cavity. Accordingly, the solutions are subject to periodic boundary conditions. The analysis reveals regions of stable stationary solutions in the model’s parameter space, and a wide range of oscillatory and chaotic behaviors. Close to the first bifurcation destabilizing the spatially uniform solution, a stationary single-humped solution is found in an asymptotic analytical form, which turns out to be in very good agreement with the numerical results. Simulations reveal a series of stable stationary multi-humped solutions.