Phase-Diffusion Equations for the Anisotropic Complex Ginzburg–Landau Equation

Equations of the Ginzburg–Landau type play a fundamental role as generic amplitude equations to which physical, chemical, biological as well as engineering systems can be reduced to describe pattern forming processes near instabilities. The subject of this paper is the 2D Anisotropic Complex Ginzburg–Landau Equation (ACGLE) and its reduction to phase equations near Benjamin–Feir–Newell (BFN) instabilities. Using a plane wave ansatz, we compute amplitude and frequency of traveling plane wave solutions of the ACGLE. The stability of these solutions is analyzed by computing the growth rate of plane wave perturbations, with small perturbational wavenumbers, from the linearized ACGLE. This allows us to determine regions in the ACGLE parameter space where the traveling plane waves are stable, partly unstable (depending on the direction of the perturbation) or fully unstable (against all long-wave perturbations). We then identify two BFN-stability boundaries in parameter space separating regions with stable traveling plane wave solutions from regions where the traveling plane wave solutions are partly unstable in one of the two directions. Codimension-1 (-2) bifurcations occur if one (both) of these conditions are met. Each of these two bifurcations is studied by applying a multiple scale expansion to derive an anisotropic Kuramoto–Sivashinsky-type (KS-type) phase equation for the phase of an ACGLE amplitude with the modulus being slaved by the phase. This enables us to generate numerical solutions of the ACGLE from numerical solutions of the underlying anisotropic KS equation. Numerical simulations near the two types of instabilities using a Fourier Galerkin method show good agreement between solutions computed directly from the ACGLE and solutions created from solutions of the phase equations. These results are important as they provide novel insights into the effect of anisotropy on phase chaotic dynamics, as well as a new numerical method to simulate the ACGLE using phase equations that gives considerable computational time savings. In addition, we apply numerical simulations to the ACGLE for initial conditions where the slaving of the modulus by the phase breaks down. In these simulations, we observe an anisotropic version of spiral defect chaos and a novel defect chaotic pattern dynamics that does not occur in isotropic systems.