Abstract
The Nikolaevskiy equation has been proposed as a model for seismic waves, electroconvection, and weak turbulence; we show that it can also be used to model transverse instabilities of fronts. This equation possesses a large-scale "Goldstone" mode that significantly influences the stability of spatially periodic steady solutions; indeed, all such solutions are unstable at onset, and the equation exhibits spatiotemporal chaos. In many applications, a weak damping of this neutral mode will be present, and we study the influence of this damping on solutions to the Nikolaevskiy equation. We examine the transition to the usual Eckhaus instability as the damping of the large-scale mode is increased, through numerical calculation and weakly nonlinear analysis. The latter is accomplished using asymptotically consistent systems of coupled amplitude equations. We find that there is a critical value of the damping below which (for a given value of the supercriticality parameter) all periodic steady states are unstable. The last solutions to lose stability lie in a cusp close to the left-hand side of the marginal stability curve.
Highlights
The Nikolaevskiy equation has been proposed as a model for seismic waves, electroconvection, and weak turbulence; we show that it can be used to model transverse instabilities of fronts
We find that there is a critical value of the damping below whichfor a given value of the supercriticality parameterall periodic steady states are unstable
The Nikolaevskiy equation is an important model of a wide range of physical systems, including certain convection problems, phase instabilities, and transverse instabilities of fronts
Summary
Has been widely studied, because of its application to several physical systems and its interesting nonlinear dynamics. The Nikolaevskiy equation exhibits a form of chaotic dynamics arising from the interaction between a pattern of finite wave number, which appears for r Ͼ 0, and a long-wave neutralor “Goldstone”͒ mode. Fujisaka et al ͓11͔ extended2͒ to two spatial dimensions and derived the corresponding amplitude equations, under the assumption that this scaling holds in that case They found it necessary to include an additional higherorder viscosity term to stabilize the high-wave-number modes in order for their simulations of their two-dimensional amplitude equations to remain finite. IV we compute numerically the steady spatially periodic “roll” solutions of the damped equation and compute their secondary stability This numerical calculation shows that there is an additional pocket of stable rolls, beyond those considered by Tribelsky16͔; this pocket is significant in containing the last rolls to remain stable as the damping coefficient is reduced. We demonstrate the subcritical onset of instability in some cases, including the no-damping case, consistent with the observed sudden onset of the instability in numerical simulations of1͒
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