# Dresden 2006 – wissenschaftliches Programm

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# SYSS: Structure Formation and Self-Organization in non-equilibrium Systems

## SYSS 4: Structure Formation and Self-Organization in non-equilibrium Systems III

### SYSS 4.2: Hauptvortrag

### Freitag, 31. März 2006, 10:45–11:15, HSZ 04

**Coarsening versus lengthscale persistence in nonequilibrium pattern-forming systems** — •Chaouqi Misbah^{1} and Paolo Politi^{2} — ^{1}Laboratoire de Spectrométrie Physique, CNRS, Univ. J. Fourier, Grenoble 1, BP87, F-38402 Saint Martin d’Hères, France — ^{2}Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, Via Madonna del Piano 10, 50019 Sesto Fiorentino, Italy

Global evolution of nonequlibrium pattern-formming systems can be
broadly classified into two important classes: (i) those which
present a persistent length scale, (ii) those which undergo a
perpetual coarsening. A general criterion about coarsening for a
class of nonlinear evolution equations describing one dimensional
pattern-forming systems will be presented. 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 λ′(*A*)>0 while a lengthscale selection
prevails if λ′(*A*)<0, where λ is the wavelength of
the pattern, and *A* the amplitude of the profile (prime refers to
differentiation). This is established thanks to the analysis of the
phase diffusion equation of the pattern. The phase diffusion
coefficient (which carries a kinetic information)is connected to
λ′(*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 a
class of equations. Another result which emerges from this study is
that the exploitation of the phase diffusion equation enables us to
determine in a rather straightforward manner the dynamical
coarsening exponent. Our calculation is exemplified on several
nonlinear equations, showing that the exact exponent is captured.
Contrary to many situations where the one dimensional character has
proven essential for the derivation of the coarsening exponent, the
present idea can be used, in principle, at any dimension. Some
speculations about the extension of the present results will be
outlined.