Secondary instabilities in the stabilized Kuramoto-Sivashinsky equation

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Abstract
The Kuramoto-Sivashinsky (KS) equation is one of the simplest but generic nonlinear equations in dissipative systems, such as hydrodynamics and moving interfaces. The KS equation with a linear stabilizing term occurs in many situations: (i) directional solidification where kinetics are decisive and (ii) terrace-edge evolution in step-flow growth in the presence of step-step interaction. The first focus is to show the genericity of the KS equation. We then present an extensive analytical and numerical study of the stabilized KS equation. It is found that this equation reveals a variety of secondary bifurcations. Besides the usual Eckhaus instability, the cellular structure exhibits (i) a period-halving of the cellular state, (ii) parity breaking (PB), (iii) vacillating breathing (VB), and (iv) oscillation with a spatial wavelength ‘‘irrationally’’ related to the basic one. Among many other features, this equation manifests also a complex mixture of PB and VB, and pairs of anomalous cells, which are observed in many experimental situations. The occurrence of some of the secondary bifurcations (e.g., PB) is examined analytically in the vicinity of the codimension-2 bifurcation where the two modes q and 2q (q being the wave number) bifurcate, and for others (e.g., VB) by analogy with the problem of a quasifree electron in a crystal. Among other results reported here, we show that the VB mode is associated with the appearance of a wave-vector gap, due to a resonance between the ‘‘incident’’ wave and the ‘‘transmitted’’ one, while the analog of the Bragg resonance is not crucial. The analytical part of our investigation is supported by the full numerical calculation.