Modulated amplitude waves in the cubic-quintic Ginzburg-Landau equation

In this paper, we begin to develop a theoretical framework for analyzing the strongly amplitude modulated numerical pulse solutions recently observed in the complex Ginzburg-Landau Equation, which is a canonical model for dissipative, weakly nonlinear systems. As such, the article also reviews background concepts of relevance to coherent structures in general dissipative systems (in regimes where such structures are stable and dominate the dynamics). This framework allows a comprehensive analysis of various bifurcations leading to transitions from one type of coherent structure to another as the system parameters are varied. It will also form a basis for future theoretical analysis of the great diversity of numerically-observed solutions, including even the spatially coherent structures with temporally quasiperiodic or chaotic envelopes observed in recent simulations.