Finite-wavelength instability coupled to a Goldstone mode: the Nikolaevskiy equation

Abstract

The Nikolaevskiy equation is considered as a simple model exhibiting spatiotemporal chaos due to the coupling of finite-wavelength patterns to a long-wavelength mode (Goldstone mode). It was originally proposed as a model for seismic waves and is also considered as a model for various physical phenomena, including electroconvection, reaction-diffusion systems and transverse instabilities of travelling fronts for chemical reactions. This equation has attracted the attention of several researchers due to its rich dynamical properties and physical applications. We are interested in studying this equation closely by means of numerical computations and asymptotic analysis. In this thesis we reinstate the dispersive terms, in contrast to most research regarding the Nikolaevskiy equation, and study the effect on the stability of spatially periodic solutions, which take the form of travelling waves. It is shown that dispersion can stabilise the travelling wave solutions, which emerge at the onset of instability of the spatially uniform state. The secondary stability plots exhibit high sensitivity on the degree of dispersion and can sometimes be remarkably complicated. Dispersive amplitude equations are derived: numerical simulations manifest behaviour similar to the non-dispersive case but there is a drift of the pattern with a certain speed. Another aspect of this thesis is analysing systems similar to the Nikolaevskiy equation, where they incorporate a Goldstone mode and possess the same symmetries. We conclude that such systems share with the Nikolaevskiy equation the fact that roll solutions are unstable at the onset of instability. We also study the amplitude equations of these systems numerically and deduce that statistical measures of their solutions depend on the ratio of the curvatures of the dispersion relation near the finite-wavelength and long-wavelength modes. Finally, we consider a system coupling a Swift-Hohenberg equation to a large-scale mode. The result of this study shows that there can be stable stationary wave solutions, in contrast to the Nikolaevskiy equation