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Universality of Crystallographic Pinning

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Abstract

We study traveling waves for reaction diffusion equations on the spatially discrete domain \({\mathbb Z^2}\). The phenomenon of crystallographic pinning occurs when traveling waves become pinned in certain directions despite moving with non-zero wave speed in nearby directions. In [19] it was shown that crystallographic pinning occurs for all rational directions, so long as the nonlinearity is close to the sawtooth, which itself was considered in [6]. 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.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Boston University, Boston, MA, 02215, USA

    Aaron Hoffman

  2. Division of Applied Mathematics, Brown University, Providence, RI, 02912, USA

    John Mallet-Paret

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  1. Aaron Hoffman
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  2. John Mallet-Paret
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Correspondence to John Mallet-Paret.

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Dedicated to Professor Jack K. Hale on the occasion of his 80th birthday.

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Hoffman, A., Mallet-Paret, J. Universality of Crystallographic Pinning. J Dyn Diff Equat 22, 79–119 (2010). https://doi.org/10.1007/s10884-010-9157-2

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  • DOI: https://doi.org/10.1007/s10884-010-9157-2

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