We analyze theoretically the dynamics of aeolian sand ripples. In order to put the study in the context we first review existing models. This paper is a continuation of two previous papers [3,4], the first one is based on symmetries and the second on a hydrodynamical model. We show how the hydrodynamical model may be modified to recover the missing terms that are dictated by symmetries. The symmetry and conservation arguments are powerful in that the form of the equation is model-independent. We then present an extensive numerical and analytical analysis of the generic sand ripple equation. We find that at the initial stage the wavelength of the ripple is that corresponding to the linearly most dangerous mode. At later stages the profile undergoes a coarsening process leading to a significant increase of the wavelength. We find that including the next higher order nonlinear term in the equation, leads naturally to a saturation of the local slope. We analyze both analytically and numerically the coarsening stage, in terms of a dynamical exponent for the mean wavelength increase. We discuss some future lines of investigations.

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A simple self-driven lattice-gas model for collective biological motion is introduced. We find weakly first order phase transition from individual random walks to collective migration. A meanfield theory is presented to support the numerical results.

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