Phase-Diffusion Equations for the Anisotropic Complex Ginzburg-Landau Equation
Derek Handwerk, Gerhard Dangelmayr, Iuliana Oprea, Patrick D. Shipman
TL;DR
This paper derives phase-diffusion equations for the anisotropic complex Ginzburg-Landau equation, analyzing stability conditions and bifurcations, and connects phase dynamics to pattern formation in anisotropic systems.
Contribution
It introduces anisotropic Kuramoto-Sivashinsky-type equations for phase dynamics and explores instability conditions and bifurcations in the ACGLE.
Findings
Identified two key instability conditions in parameter space.
Derived phase equations governing pattern modulations.
Linked phase solutions to the original ACGLE solutions.
Abstract
The anisotropic complex Ginzburg-Landau equation (ACGLE) describes slow modulations of patterns in anisotropic spatially extended systems near oscillatory (Hopf) instabilities with zero wavenumbers. Traveling wave solutions to the ACGLE become unstable near Benjamin-Feir-Newell instabilities. We determine two instability conditions in parameter space and study codimension-one (-two) bifurcations that occur if one (two) of the conditions is (are) met. We derive anisotropic Kuramoto-Sivashinsky-type equations that govern the phase of the complex solutions to the ACGLE and generate solutions to the ACGLE from solutions of the phase equations.
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Taxonomy
TopicsNonlinear Dynamics and Pattern Formation · Plant Reproductive Biology · Nonlinear Photonic Systems