Nonlinear dynamics in one dimension: On a criterion for coarsening and its temporal law

TL;DR

This paper introduces a criterion based on steady state solutions to predict coarsening in one-dimensional nonlinear pattern-forming systems, linking pattern wavelength behavior to coarsening dynamics and providing a way to determine coarsening exponents.

Contribution

It establishes a novel analytical criterion for coarsening based solely on steady state solution branches, connecting kinetics and pattern properties in nonlinear evolution equations.

Findings

Coarsening occurs if lambda'(A)>0, otherwise length scale is fixed.

The phase diffusion coefficient D(lambda) relates to steady state properties.

The method accurately predicts coarsening exponents in tested equations.

Abstract

We develop a general criterion about coarsening for a class of nonlinear evolution equations describing one dimensional pattern-forming systems. This criterion allows one to discriminate between the situation where a coarsening process takes place and the one where the wavelength is fixed in the course of time. An intermediate scenario may occur, namely `interrupted coarsening'. The power of the criterion lies in the fact that the statement about the occurrence of coarsening, or selection of a length scale, can be made by only inspecting the behavior of the branch of steady state periodic solutions. The criterion states that coarsening occurs if lambda'(A)>0 while a length scale selection prevails if lambda'(A)<0, where $lambda$ is the wavelength of the pattern and A is the amplitude of the profile. This criterion is established thanks to the analysis of the phase diffusion equation of the pattern. We connect the phase diffusion coefficient D(lambda) (which carries a kinetic information) to lambda'(A), which refers to a pure steady state property. The relationship between kinetics and the behavior of the branch of steady state solutions is established fully analytically for several classes of equations. Another important and new result which emerges here is that the exploitation of the phase diffusion coefficient enables us to determine in a rather straightforward manner the dynamical coarsening exponent. Our calculation, based on the idea that |D(lambda)|=lambda^2/t, is exemplified on several nonlinear equations, showing that the exact exponent is captured. Some speculations about the extension of the present results to higher dimension are outlined.