Homoclinic solutions for fourth order traveling wave equations

TL;DR

This paper proves the existence of homoclinic solutions for specific fourth order traveling wave equations with two different potential types, advancing understanding of these solutions in nonlinear differential equations.

Contribution

It establishes the existence of homoclinic solutions for all relevant parameters in two classes of fourth order equations, partially confirming a conjecture by Chen and McKenna.

Findings

Homoclinic solutions exist for all β in (0, β*) for suspension bridge type.

Homoclinic solutions exist for all β in (0, β*) for Swift-Hohenberg type.

Partially solves Chen–McKenna conjecture on homoclinic solutions.

Abstract

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}.