Notes on homoclinic solutions of the steady Swift-Hohenberg equation

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 ....

Keywords

Homoclinic solutions / Normal form / Reversibility

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