Homoclinic Solutions for Fourth Order Traveling Wave Equations

Abstract

We consider homoclinic solutions of fourth order equations $u^{””} + \beta^2 u^{”} + V_u(u) = 0$ in $\mathbb{R}$, where $V(u)$ is either the suspension bridge type $V(u) = e^u-1-u$ or the Swift–Hohenberg type $V(u) = \frac{1}{4}(u^2-1)^2$. For the suspension bridge type equation, we prove existence of a homoclinic solution for all $\beta \in (0, \beta_{*})$, where $\beta_{*} = 0.7427\cdots$. For the Swift–Hohenberg type equation, we prove existence of a homoclinic solution for each $\beta \in (0, \beta_{*})$, where $\beta_{*} = 0.9342\cdots$. This partially solves a conjecture of Chen–McKenna [Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 355 (1997), pp. 2175–2184].