Fronts, pulses, sources and sinks in generalized complex Ginzburg-Landau equations

Abstract

An important element in the long-time dynamics of pattern forming systems is a class of solutions we will call “coherent structures”. These are states that are either themselves localized, or that consist of domains of regular patterns connected by localized defects or interfaces. This paper summarizes and extends recent work on such coherent structures in the one-dimensional complex Ginzburg-Landau equation and its generalizations, for which rather complete information can be obtained on the existence and competition of fronts, pulses, sources and sinks. For the special subclass of uniformly translating structures, the solutions are derived from a set of ordinary differential equations that can be interpreted as a flow in a three-dimensional phase space. Fixed points of the flow correspond to the two basic building blocks of coherent structures, uniform amplitude states and evanescent waves whose amplitude decreases smoothly to zero. A study of the stability of the fixed points under the flow leads to results on the existence and multiplicity of the different coherent structures. The dynamical analysis of the original partial differential equation focusses on the competition between pulses and fronts, and is expressed in terms of a set of conjectures for front propagation that generalize the “marginal stability” and “pinch-point” approaches of earlier authors. These rules, together with an exact front solution whose dynamics plays an important role in the selection of patterns, yield an analytic expression for the upper limit of the range of existence of pulse solutions, as well as a determination of the regions of parameter space where uniformly translating front solutions can exist. Extensive numerical simulations show consistency with these rules and conjectures for the existence of fronts and pulses. In the parameter ranges where no uniformly translating fronts can exist, examples are shown of irregularly spreading fronts that generate strongly chaotic regions, as well as nonuniformly translating fronts that lead to uniform amplitude states. Recent perturbative treatments based on expansions about the nonlinear Schrödinger equation are generalized to perturbations of the cubic-quintic and derivative Schrödinger equations, for which both pulses and fronts exist in the unperturbed system. Comparison of the results with the exact solutions shows that the perturbation theory only yields a subset of the relevant solutions. Nevertheless, those that are obtained are found to be consistent with the general conjectures, and in particular they provide an analytic demonstration of front/pulse competition. While the discussion of the competition between fronts and pulses focusses on the complex Ginzburg-Landau equation with quintic terms and a subcritical bifurcation, a number of results are also presented for the cubic equation. In particular, the existence of a family of moving source solutions derived by Bekki and Nozaki for this equation contradicts the naive counting arguments. We attribute this contradiction to a hidden symmetry of the solution but have not been able to show explicitly how this symmetry affects the phase space orbits.