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Notes on homoclinic solutions of the steady Swift-Hohenberg equation

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Abstract

This paper considers the steady Swift-Hohenberg equation

$$u'''' + \beta ^2 u'' + u^3 - u = 0.$$

Using the dynamic approach, the authors prove that it has a homoclinic solution for each \(\beta \in \left[ {\sqrt[4]{8} - \varepsilon _0 ,\sqrt[4]{8}} \right)\), where ε 0 is a small positive constant. This slightly complements Santra and Wei’s result [Santra, S. and Wei, J., Homoclinic solutions for fourth order traveling wave equations, SIAM J. Math. Anal., 41, 2009, 2038–2056], which stated that it admits a homoclinic solution for each β ∈ (0, β 0) where β 0 = 0.9342 ....

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Author information

Correspondence to Shengfu Deng.

Additional information

Project supported by the PhD Start-up Fund of the Natural Science Foundation of Guangdong Province (No. S2011040000464), the Project of Department of Education of Guangdong Province (No. 2012KJCX0074), the China Postdoctoral Science Foundation-Special Project (No. 201104077), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry (No. (2012)940), the Natural Fund of Zhanjiang Normal University (No. LZL1101) and the Doctoral Project of Zhanjiang Normal University (No. ZL1109).

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Deng, S., Guo, B. & Li, X. Notes on homoclinic solutions of the steady Swift-Hohenberg equation. Chin. Ann. Math. Ser. B 34, 917–920 (2013). https://doi.org/10.1007/s11401-013-0801-0

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Keywords

  • Homoclinic solutions
  • Normal form
  • Reversibility

2000 MR Subject Classification

  • 34B60
  • 34C37
  • 35B32
  • 37C29