Eur. Phys. J. B 83, 29-37 (2011)
https://doi.org/10.1140/epjb/e2011-20403-8

Exact steady states to a nonlinear surface growth model

¹ LAMFA, CNRS UMR 6140, Department of Mathematics Université de Picardie Jules Verne, 33, rue, Saint-Leu Amiens, France
² LPMC, Department of Physic, Université de Picardie Jules Verne, 33, rue saint-Leu, Amiens, France
³ LIPhy, CNRS UMR 5588, Université Joseph Fourier, 140 Avenue de la Physique, Saint Martin d'Hères, France

Corresponding author: ^a guedda@u-picardie.fr

Received: 29 May 2011
Revised: 10 July 2011
Published online: 12 September 2011

Abstract

We report on exact stationary solutions to a nonlinear evolution equation describing the collective step meander on a vicinal surface subject to the Bales-Zangwill growth instability [O. Pierre-Louis et al., Phys. Rev. Lett. 80, 4221 (1998)]. Firstly, attention is focused on periodic solutions (steady states) which admit vertical points (or diverging local slopes). Such solutions, which are determined by a theoretical analysis, reveal that the nonlinear evolution equation may admit a non stationary solution with spike singularities or/and caps (dead-core solution) at maxima or/and minima. In a second part, steady states are, mathematically, generalized to a family of evolution equations. Finally, the effect of smoothening by step-edge diffusion is also revisited.