Homoclinic Solutions for Fourth Order Traveling Wave Equations

We consider homoclinic solutions of fourth order equations u ′′′′ + βu ′′ + Vu(u) = 0 in R , where V (u) is either the suspension bridge type V (u) = eu − 1 − u or SwiftHohenberg type V (u) = 1 4 (u2 − 1)2. For the suspension bridge type equation, we prove existence of a homoclinic solution for all β ∈ (0, β∗) where β∗ = 0.7427 · · · . For the Swift-Hohenberg type equation, we prove existence of a homoclinic solution for each β ∈ (0, β∗), where β∗ = 0.9342 · · · . This partially solves a conjecture of Chen–McKenna [12].