Homoclinic solutions for fourth order traveling wave equations

Abstract

We consider homoclinic solutions of fourth order equations $$u^{""}+{\beta }^2{u}^{"}+V_u(u)=0in\mathbb{R},$$ where $V(u)$ is either the suspension bridge type $V(u)=e^u-1-u$ or Swift-Hohenberg type $V(u)=1/4(u^2-1{)}^2$. For the suspension bridge type equation, we prove existence of a homoclinic solution for {\em all} $\beta \in (0,{\beta }_{\ast })$ where ${\beta }_{\ast }=\mathrm{0.7427...}$. For the Swift-Hohenberg type equation, we prove existence of a homoclinic solution for each $\beta \in (0,{\beta }_{\ast })$, where ${\beta }_{\ast }=\mathrm{0.9342...}$. This partially solves a conjecture of Chen--McKenna \cite{YCM1}.

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