Instabilities and Dynamics of Weakly Subcritical Patterns

Abstract

The bifurcation to one-dimensional weakly subcritical periodic patterns is described by the cubic-quintic Ginzburg-Landau equation At = µA + Axx + i (a 1 |A |2 Ax + a 2 A 2 Ax * ) + b |A |2 A - |A |4 A . These periodic patterns may in turn become unstable through one of two different mechanisms, an Eckhaus instability or an oscillatory instability. We study the dynamics near the instability threshold in each of these cases using the corresponding modulation equations and compare the results with those obtained from direct numerical simulation of the equation. We also study the stability properties and dynamical evolution of different types of fronts present in the protosnaking region of this equation. The results provide new predictions for the dynamical properties of generic systems in the weakly subcritical regime.