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Dimension Breaking from Spatially-Periodic Patterns to KdV Planforms

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Abstract

The problem of dimension breaking, for gradient elliptic partial differential equations in the plane, from a family of one-dimensional spatially periodic patterns (rolls) is considered. Conditions on the family of rolls are determined that lead to dimension breaking in the plane governed by a KdV equation relative to the periodic state. Since the KdV equation is time-independent, the \(N\)-pulse solutions of KdV provide a sequence of multi-pulse planforms in the plane bifurcating from the rolls. The principal examples are the nonlinear Schrödinger equation, with evolution in the plane, and the steady Swift–Hohenberg equation with weak transverse variation.

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Acknowledgments

Helpful discussions with Sergey Zelik are gratefully acknowledged.

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Authors and Affiliations

  1. Department of Mathematics, University of Surrey, Guildford, GU2 7XH, England, UK

    Thomas J. Bridges

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  1. Thomas J. Bridges
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Correspondence to Thomas J. Bridges.

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Dedicated to the memory of Klaus Kirchgässner.

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Bridges, T.J. Dimension Breaking from Spatially-Periodic Patterns to KdV Planforms. J Dyn Diff Equat 27, 443–456 (2015). https://doi.org/10.1007/s10884-014-9405-y

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  • DOI: https://doi.org/10.1007/s10884-014-9405-y

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