High-energy solutions for a phase transition problem

In this article we study the existence and profile of high-energy solutions for the phase transition model ε2u″−a(x)W′(u)=0, in (−1,1),u′(−1)=u′(1)=0,with W a double well balanced potential and ε is a small parameter. These solutions will be shown to be highly oscillatory. We will characterize their asymptotic behavior in terms of an envelope function.