Transverse modulational dynamics of quenched patterns

TL;DR

This paper derives a one-dimensional viscous Burgers' equation to describe the transverse modulational dynamics of striped patterns formed behind a moving quench, providing insights into defect behavior and pattern selection.

Contribution

It introduces a multiple-scale analysis linking quench-induced pattern control to a Burgers' equation for transverse dynamics in complex Ginzburg-Landau and Swift-Hohenberg models.

Findings

Burgers' equation captures defect dynamics accurately

Wavenumber selection influences nonlinear flux

Transverse diffusivity determines viscosity parameter

Abstract

We study the modulational dynamics of striped patterns formed in the wake of a planar directional quench. Such quenches, which move across a medium and nucleate pattern-forming instabilities in their wake, have been shown in numerous applications to control and select the wavenumber and orientation of striped phases. In the context of the prototypical complex Ginzburg-Landau and Swift-Hohenberg equations, we use a multiple-scale analysis to derive a one-dimensional viscous Burgers' equation which describes the long-wavelength modulational and defect dynamics in the direction transverse to the quenching motion, that is along the quenching line. We show that the wavenumber selecting properties of the quench determines the nonlinear flux parameter in the Burgers' modulation equation, while the viscosity parameter of the Burgers' equation is naturally determined by the transverse diffusivity of the pure stripe state. We use this approximation to accurately characterize the transverse dynamics of several types of defects formed in the wake, including grain boundaries and phase-slips.