Patterns produced by a short-wave instability in the presence of a zero mode

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Abstract
One-dimensional pattern-forming systems with a quadratic nonlinearity and a long-wave zero mode in the governing equations are considered. The pattern-generating instability, which may be both steady and oscillatory, is assumed to set in at a finite wave number. Physical examples are the instability of a front of the laser-sustained evaporation of a solid and the instability of seismic waves in a viscoelastic medium. A system of generalized Ginzburg-Landau equations for the complex envelope of a fundamental harmonic and for the real slowly relaxing zero mode are derived in a general form [for the case of the steady instability, essentially the same equations have been proposed earlier by Coullet and Fauve, Phys. Rev. Lett. 55, 2857 (1985)]. Using these equations, stability of spatially periodic patterns is investigated. The main result is that the stability band is anomalously narrow in comparison with the classical Eckhaus band. For the case of the steady instability, the evolution of long-wave modulations of the patterns is investigated in the geometric-optics approximation. It is demonstrated that, unlike the usual Ginzburg-Landau equation, the equations derived in the present work allow for modulation profiles of a permanent shape. The profiles may be transient layers moving at a constant velocity, or quiescent periodic ‘‘superstructures,’’ including a ‘‘soliton’’ as a limiting case. At least some of those profiles can be stable.

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