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

We develop a general criterion about coarsening for some classes 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 on which a brief account has been given [Politi and Misbah, Phys. Rev. Lett. 92, 090601 (2004)], and which we extend here to more general equations, 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 ${\lambda }^{\prime }\left(A\right)>0$ while a length scale selection prevails if ${\lambda }^{\prime }\left(A\right)<0$, where $\lambda$ is the wavelength of the pattern and $A$ is the amplitude of the profile (prime refers to differentiation). This criterion is established thanks to the analysis of the phase diffusion equation of the pattern. We connect the phase diffusion coefficient $D\left(\lambda \right)$ (which carries a kinetic information) to ${\lambda }^{\prime }\left(A\right)$, 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. 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 $\mid D\left(\lambda \right)\mid \sim {\lambda }^2∕t$, is exemplified on several nonlinear equations, showing that the exact exponent is captured. We are not aware of another method that so systematically provides the coarsening exponent. Contrary to many situations where the one-dimensional character has proven essential for the derivation of the coarsening exponent, this idea can be used, in principle, at any dimension. Some speculations about the extension of the present results are outlined.