Oscillatory pattern formation with a conserved quantity

Abstract

The influence of a conserved quantity on an oscillatory pattern-forming instability is examined in one space dimension. Amplitude equations are derived which are not only generic for systems with a pseudoscalar conserved quantity (e.g. rotating convection, magnetoconvection) but also applicable to systems with a scalar conserved quantity. The stability properties of both travelling and standing waves are analysed, with particular progress being possible in the limit of long-wavelength perturbations. For both forms of waves, the corresponding modulational stability boundaries are significantly altered by the presence of the conserved quantity; also, new instabilities are generated. For general perturbations, the full stability regions are found numerically. Simulations of the nonlinear governing equations are performed using a pseudo-spectral code; a variety of stable attractors are thus found of varying degrees of complexity. Previously unseen, highly localized, solutions are observed.