Weakly versus highly nonlinear dynamics in 1 D systems

– 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. In the study of pattern formation, weakly nonlinear equations play a central role. By construction, these equations catch the main effects of nonlinearities via a limited number of nonlinear terms added to a linear equation. This approach has been used for a wide variety of physical systems such as crystal growth [1], reaction-diffusion systems [2], flame fronts [3], or phase separation [4]. Weakly nonlinear equations can be derived from a multi-scale analysis when separation of scales is possible. This is for example the case in the vicinity of an instability threshold, where the system is weakly unstable, or in the analysis of amplitude and phase dynamics of modulated structures [5]. These equations are also obtained from renormalization techniques [6]. Some analysis and attempt of classification of generic nonlinear equations based on symmetry or geometry have already been reported in the literature [5, 7, 8]. The most systematic approach up to now was that of Ref. [8], where nonlinear equations result from the expansion in Cartesian coordinates of dynamics expressed in intrinsic coordinates. We here present a more general approach based on a multi-scale analysis. For the sake of simplicity, we assume that dynamics is local and that an instability appears at long wavelength at the instability threshold. From the assumption that the stabilizing or nonlinear terms do not scale with the small parameter of the expansion ǫ, we find that the Benney, and the sand ripple [9] equations are expected in systems with translational invariance, and that the convective Ginzburg-Landau equation is expected in absence of translational invariance. Furthermore, our approach determines the range of validity of weakly nonlinear expansions even when ǫ is present in the stabilizing or nonlinear terms. As a central result, we show that the weakly nonlinear approach breaks down for a large class of front dynamics. The main c EDP Sciences