Instability of spatially quasi-periodic states of the Ginzburg-Landau equation

Abstract

The Ginzburg-Landau (GL) equation with real coefficients is a model equation appearing in superconductor physics and near-critical hydrodynamic stability problems. The stationary GL equation has a two-parameter ( I 1 , I 2 ) family of spatially quasi-periodic (QP) states with frequencies ( ω 1 , ω 2 ) and frequency map with determinant ∆ K = ∂( ω 1 , ω 2 ) / ∂( I 1 , I 2 ). In this paper the linear stability of these QP states is studied and an expression for the stability exponent is obtained which has a novel geometric interpretation in terms of ∆ K : when ∆ K < 0 the spatially QP state is unstable and ∆ K > 0 is a necessary but not sufficient condition for linear stability. There is an interesting relation between ∆ K and the KAM persistence theorem for invariant toroids.