Explicit stationary solutions in multiple well dynamics and non-uniqueness of interfacial energy densities

Abstract

We present a theory that enables us to construct heteroclinic connections in closed form for $2\bf{u}_{xx}=W_{\bf u}({\bf u})$, where $x\in\mathbb{R},\;{\bf u}(x)\in \mathbb{R}^2$ and $W$ is a smooth potential with multiple global minima. In particular, multiple connections between global minima are constructed for a class of potentials. With these potentials, numerical simulations for the vector Allen-Cahn equation ${\bf u}_t= 2\epsilon^2 \Delta {\bf u}-W_{\bf u}({\bf u})$ in two space dimensions with small $\epsilon>0$, show that between any fixed pair of phase regions, interfaces are partitioned into segments of different energy densities, where the proportions of the length of these segments are changing with time. Our results imply that for the case of triple-well potentials the usual Plateau angle conditions at the triple junction are generally violated.