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Bifurcation to Coherent Structures in Nonlocally Coupled Systems

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Abstract

We show bifurcation of localized spike solutions from spatially constant states in systems of nonlocally coupled equations in the whole space. The main assumptions are a generic bifurcation of saddle-node or transcritical type for spatially constant profiles, and a symmetry and second moment condition on the convolution kernel. The results extend well known results for spots, spikes, and fronts, in locally coupled systems on the real line, and for radially symmetric profiles in higher space dimensions. Rather than relying on center manifolds, we pursue a more direct approach, deriving leading order asymptotics and Newton corrections for error terms. The key ingredient is smoothness of Fourier multipliers arising from discrepancies between nonlocal operators and their local long-wavelength approximations.

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Acknowledgements

A.S. and T.T. gratefully acknowledge support under grant NSF-DMS–1612441.

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  1. School of Mathematics, University of Minnesota, 206 Church St. S.E., Minneapolis, MN, 55455, USA

    Arnd Scheel & Tianyu Tao

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  1. Arnd Scheel
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  2. Tianyu Tao
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Correspondence to Arnd Scheel.

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Dedicated to the memory of George Sell, mentor, colleague, and friend

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Scheel, A., Tao, T. Bifurcation to Coherent Structures in Nonlocally Coupled Systems. J Dyn Diff Equat 31, 1107–1127 (2019). https://doi.org/10.1007/s10884-017-9613-3

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  • DOI: https://doi.org/10.1007/s10884-017-9613-3

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