Dynamic effects induced by renormalization in anisotropic pattern forming systems

TL;DR

This paper investigates how anisotropic nonlinearities in 2D pattern-forming systems lead to unexpected dynamics, such as ripple rotation, due to differential renormalization effects, with implications for experiments.

Contribution

It demonstrates that full 2D generalizations of pattern equations can produce novel behaviors like ripple rotation caused by anisotropic nonlinear renormalization.

Findings

Ripple pattern rotates by 90 degrees under certain conditions.

Nonlinear parameter renormalization occurs at different rates in two dimensions.

Potential experimental systems can exhibit this anisotropic dynamical behavior.

Abstract

The dynamics of patterns in large two-dimensional domains remains a challenge in non-equilibrium phenomena. Often it is addressed through mild extensions of one-dimensional equations. We show that full 2D generalizations of the latter can lead to unexpected dynamical behavior. As an example we consider the anisotropic Kuramoto-Sivashinsky equation, that is a generic model of anisotropic pattern forming systems and has been derived in different instances of thin film dynamics. A rotation of a ripple pattern by $90^{\circ}$ occurs in the system evolution when nonlinearities are strongly suppressed along one direction. This effect originates in non-linear parameter renormalization at different rates in the two system dimensions, showing a dynamical interplay between scale invariance and wavelength selection. Potential experimental realizations of this phenomenon are identified.