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Diffusive Stability of Spatially Periodic Solutions of the Brusselator Model

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Abstract

Applying the Lyapunov–Schmidt reduction approach introduced by Mielke and Schneider in their analysis of the fourth-order scalar Swift–Hohenberg equation, we carry out a rigorous small-amplitude stability analysis of Turing patterns for the canonical second-order system of reaction–diffusion equations given by the Brusselator model. Our results confirm that stability is accurately predicted in the small-amplitude limit by the formal Ginzburg–Landau amplitude equations, rigorously validating the standard weakly unstable approximation and the Eckhaus criterion.

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Authors and Affiliations

  1. Indiana University, Bloomington, IN, 47405, USA

    Alim Sukhtayev, Kevin Zumbrun & Raghavendra Venkatraman

  2. Kongju National University, Gongju, Korea

    Soyeun Jung

Authors
  1. Alim Sukhtayev
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  2. Kevin Zumbrun
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  3. Soyeun Jung
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  4. Raghavendra Venkatraman
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Corresponding author

Correspondence to Kevin Zumbrun.

Additional information

Communicated by W. Schlag

Research of K.Z. was partially supported under NSF Grants Nos. DMS-0300487 and DMS-0801745. Research of S.J. was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (No. 2016R1C1B1009978). Research of R.V. was partially supported under NSF Grants Nos. DMS-1101290 and DMS-1362879.

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Sukhtayev, A., Zumbrun, K., Jung, S. et al. Diffusive Stability of Spatially Periodic Solutions of the Brusselator Model. Commun. Math. Phys. 358, 1–43 (2018). https://doi.org/10.1007/s00220-017-3056-x

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  • DOI: https://doi.org/10.1007/s00220-017-3056-x

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