Phase slips and the Eckhaus instability

Phase Slips and the Eckhaus Instability

We consider the Ginzburg-Landau equation, @ t u = @ 2 x u+u?ujuj 2 , with complex amplitude u(x; t). We first analyze the phenomenon of phase slips as a consequence of the local shape of u. We next prove a global theorem about evolution from an Eckhaus unstable state, all the way to the limiting stable finite state, for periodic perturbations of Eckhaus unstable periodic initial data. Equipped ...

Eckhaus instability and homoclinic snaking.

Homoclinic snaking is a term used to describe the back and forth oscillation of a branch of time-independent spatially localized states in a bistable, spatially reversible system as the localized structure grows in length by repeatedly adding rolls on either side. This behavior is simplest to understand within the subcritical Swift-Hohenberg equation, but is also present in the subcritical regi...

The Eckhaus instability in hexagonal patterns

The Eckhaus instability of hexagonal patterns is studied within the model of three coupled envelope equations for the underlying roll systems. The regions of instability in the parameter space are found analytically from both the phase approximation and a full system of amplitude equations. Beyond the stability limits of hexagons two different modes go unstable. Both provide symmetry breaking o...

Eckhaus instability in systems with large delay.

The dynamical behavior of various physical and biological systems under the influence of delayed feedback or coupling can be modeled by including terms with delayed arguments in the equations of motion. In particular, the case of long delay may lead to complicated and high-dimensional dynamics. We investigate the effects of delay in systems that display an oscillatory instability (Hopf bifurcat...

Pattern selection near the onset of convection: The Eckhaus instability.