Multiple-end solutions to the Allen–Cahn equation in R2
- del Pino, Manuel
- Kowalczyk, Michał
- Pacard, Frank
- Wei, Juncheng
DOIAbstract
AbstractWe construct a new class of entire solutions for the Allen–Cahn equation Δu+(1−u2)u=0, in R2(∼C). Given k⩾1, we find a family of solutions whose zero level sets are, away from a compact set, asymptotic to 2k straight lines (which we call the ends). These solutions have the property that there exist θ0<θ1<⋯<θ2k=θ0+2π such that limr→+∞u(reiθ)=(−1)j uniformly in θ on compact subsets of (θj,θj+1), for j=0,…,2k−1
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