四阶行波wave equations的同宿解
Homoclinic solutions for fourth order traveling wave equations
摘要：
考虑四阶方程$$ u^{""} + β^2 u^{"} + V_u (u)=0 {in} \R ,$$的同宿解，其中$V(u)$是吊桥型$V(u)=e^u-1-u$或Swift-Hohenberg型$ V(u)= {1/4}(u^2-1)^2$ .对于吊桥型方程，证明了{ \em all} $ β\in (0, β_*)$的同宿解的存在性，其中$ β_{*}= 0.7427...$ .对于Swift-Hohenberg型方程，证明了每个$β\in (0, β_{*})$的同宿解的存在性，其中$β_{*}=0.9342...$ .这部分地解决了Chen-McKenna的一个猜想\cite{YCM1} .
英文摘要：
We consider homoclinic solutions of fourth order equations $$ u^{""} + β^2 u^{"} + V_u (u)=0 {in} \R ,$$ 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} $ β\in (0, β_*)$ where $ β_{*}= 0.7427...$. For the Swift-Hohenberg type equation, we prove existence of a homoclinic solution for each $β\in (0, β_{*})$, where $β_{*}=0.9342...$. This partially solves a conjecture of Chen--McKenna \cite{YCM1}.
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