Bifurcation Analysis of the Eckhaus Instability

Abstract

The Eckhaus instability [1] is the only purely two-dimensional instability occurring in convection [2], Taylor-Couette flow [3], liquid crystals [4], and other pattern-forming systems. Despite its widespread occurrence, the Eckhaus instability has not been analyzed in bifurcation-theoretic terms. It can be studied analytically by averaging over the “depth” [5], leading to the classic one-dimensional Ginzburg-Landau equation:

$$ \frac{{\partial A}}{{\partial t}} = \mu A + \frac{{{\partial ^2}A}}{{\partial {x^2}}} - {\left| A \right|^2}A. $$

(1)