Pattern formation in the damped Nikolaevskiy equation

Solitary-like and modulated wavepackets in the Couette-Taylor flow with a radial temperature gradient

We have performed numerical and experimental studies of the flow in a large aspect ratio Couette-Taylor system with a rotating inner cylinder and a fixed radial temperature gradient. The base flow state is a superposition of an azimuthal flow induced by rotation and an axial large convective cell induced by the temperature gradient. For a relatively large temperature gradient, the rotation rate of the inner cylinder destabilizes the convective cell to give rise to travelling wave pattern through a subcritical bifurcation. This wave pattern is associated with a temperature mode and it consists of helical vortices travelling in the annulus. In a small range of the rotation rate, helical vortices have longitudinal meandering leading to the formation of kinks randomly distributed, leading to spatio-temporal disordered patterns. The flow becomes regular for a large interval of rotation rate. The friction, the momentum and the heat transfer coefficients are computed and found to be independent of the heating direction. This article is part of the theme issue 'Taylor-Couette and related flows on the centennial of Taylor's seminal Philosophical Transactions paper (part 1)'.

Slow diffusive structure in Nikolaevskii turbulence

Weak turbulence has been investigated in nonlinear-nonequilibrium physics to understand universal characteristics near the transition point of ordered and disordered states. Here the one-dimensional Nikolaevskii turbulence, which is a mathematical model of weak turbulence, is studied theoretically. We calculate the velocity field of the Nikolaevskii turbulence assuming a convective structure and carry out tagged-particle simulations in the flow to clarify the Nikolaevskii turbulence from the Lagrangian description. The tagged particle diffuses in the disturbed flow and the diffusion is superdiffusive in an intermediate timescale between ballistic and normal-diffusive scale. The diffusion of the slow structure is characterized by the power law for the control parameter near the transition point of the Nikolaevskii turbulence, suggesting that the diffusive characteristics of the slow structure remain scale invariant. We propose a simplified model, named two-scale Brownian motion, which reveals a hierarchy in the Nikolaevskii turbulence.

Coarsening to chaos-stabilized fronts

We investigate a model for pattern formation in the presence of Galilean symmetry proposed by Matthews and Cox [Phys. Rev. E 62, R1473 (2000)], which has the form of coupled generalized Burgers- and Ginzburg-Landau-type equations. With only the system size L as a parameter, we find distinct "small-L" and "large-L" regimes exhibiting clear differences in their dynamics and scaling behavior. The long-time statistically stationary state contains a single L-dependent front, stabilized globally by spatiotemporally chaotic dynamics confined away from the front. For sufficiently large domains, the transient dynamics include a state consisting of several viscous shocklike structures that coarsens gradually, before collapsing to a single front when one front absorbs the others.

Nikolaevskiy equation with dispersion

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 stabilize 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.

Anomalous scaling on a spatiotemporally chaotic attractor

The Nikolaevskiy model for pattern formation with continuous symmetry exhibits spatiotemporal chaos with strong scale separation. Extensive numerical investigations of the chaotic attractor reveal unexpected scaling behavior of the long-wave modes. Surprisingly, the computed amplitude and correlation time scalings are found to differ from the values obtained by asymptotically consistent multiple-scale analysis. However, when higher-order corrections are added to the leading-order theory of Matthews and Cox, the anomalous scaling is recovered.

Patterns in dissipative systems with weakly broken continuous symmetry

Patterns in dissipative systems with weakly broken symmetry are studied based upon the simplest canonical equation (generalized Nikolaevskiy model). A generic cubic dispersion equation governing stability of steady spatially periodic patterns is derived and analyzed. A domain of stable states in the space of the problem parameters (stability balloon) is obtained. It is shown that the domain is characterized by unusual scaling properties, so that its different parts obey different scalings. The results obtained may be applied to describe instabilities of advancing fronts and interfaces, pattern formation in reaction-diffusion systems, nonlinear evolution of seismic waves, and other phenomena.