Homoclinic solutions for fourth order traveling wave equations

We consider homoclinic solutions of fourth order equations $$u^{""} + \beta^2 u^{"} + V_u (u)=0 {in} \R ,$$ 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} $\beta \in (0, \beta_*)$ where $\beta_{*}= 0.7427...$. For the Swift-Hohenberg type equation, we prove existence of a homoclinic solution for each $\beta \in (0, \beta_{*})$, where $\beta_{*}=0.9342...$. This partially solves a conjecture of Chen--McKenna \cite{YCM1}.