The Nikolaevskiy equation with dispersion

TL;DR

This paper investigates how adding dispersive terms to the Nikolaevskiy equation influences the stability of traveling waves, revealing complex stability patterns and transitions to chaos in systems modeling seismic waves and electroconvection.

Contribution

It demonstrates that dispersive terms can stabilize certain traveling waves and explores the resulting complex stability and chaos transitions.

Findings

Dispersive terms can stabilize some traveling waves.

The secondary stability diagram is highly complex.

Transition to chaos is influenced by dispersion effects.

Abstract

The Nikolaevskiy equation was originally proposed as a model for seismic waves and is also a model for a wide variety of systems incorporating a neutral, Goldstone mode, including electroconvection and reaction-diffusion systems. It is known to exhibit chaotic dynamics at the onset of pattern formation, at least when the dispersive terms in the equation are suppressed, as is commonly the practice in previous analyses. In this paper, the effects of reinstating the dispersive terms are examined. It is shown that such terms can stabilise some of the spatially periodic traveling waves; this allows us to study the loss of stability and transition to chaos of the waves. The secondary stability diagram (Busse balloon) for the traveling waves can be remarkably complicated.