Weakly vs. highly nonlinear dynamics in 1D systemsO. Pierre-Louis
CNRS/Laboratoire de Spectrométrie Physique, Université Joseph Fourier Grenoble 1 BP 87, F-38402 St Martin d'Hères, France
received 19 August 2005; accepted in final form 19 October 2005
published online 16 November 2005
We analyze the morphological transition of a one-dimensional system described by a scalar field, where a flat state looses its stability. This scalar field may for example account for the position of a crystal growth front, an order parameter, or a concentration profile. We show that two types of dynamics occur around the transition: weakly nonlinear dynamics, or highly nonlinear dynamics. The conditions under which highly nonlinear evolution equations appear are determined, and their generic form is derived. Finally, examples are discussed.
05.45.-a - Nonlinear dynamics and nonlinear dynamical systems.
05.70.Ln - Nonequilibrium and irreversible thermodynamics.
47.54.+r - Pattern selection; pattern formation.
© EDP Sciences 2005