2008/11/01 by A. Hoffman, John Mallet‐Paret, Hoffman, A. +2
Computer Science · Mathematics · Physics and Astronomy · #34C37 #35K57 #37C99 #37L60 #Dynamical Systems (math.DS) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Quantum chaos and dynamical systems #Theoretical and Computational Physics #math.DS #msc:34C37 #msc:35K57 #msc:37C99 #msc:37L60
paper · pdf · doi:10.48550/arxiv.0811.0093
55 pages
arxiv created 2008/11/01 · openalex publication_date 2008/11/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
We study traveling waves for reaction diffusion equations on the spatially discrete domain \Z2. The phenomenon of crystallographic pinning occurs when traveling waves become pinned in certain directions despite moving with non-zero wave speed in nearby directions. Mallet-Paret has shown that crystallographic pinning occurs for all rational directions, so long as the nonlinearity is close to the sawtooth. In this paper we show that crystallographic pinning holds in the horizontal and vertical directions for bistable nonlinearities which satisfy a specific computable generic condition. The proof is based on dynamical systems. In particular, it relies on an examination of the heteroclinic chains which occur as singular limits of wave profiles on the boundary of the pinning region.