Dynamics of curved travelling fronts for the discrete Allen-Cahn equation on a two-dimensional lattice

Figure 1.  In §5 we show that for each $ j\in \mathbb{Z} $ and $ t\gg 0 $, the function $ i\mapsto u_{i,j}(t) $ is monotonic inside an interfacial region $ I $ that is depicted in light blue. The dark blue dots represent the horizontal solution slice $ i\mapsto u_{i,j}(t) $. Since $ u $ is monotonic inside $ I $, we can find an unique value $ i_* $ for which $ u_{i_*, j}(t) \leq 1/2 < u_{i_*+1, j}(t) $. We subsequently shift the travelling wave profile $ \Phi $ in such a way that it matches the solution slice at $ i_* $. The phase $ \gamma_j(t) $ is then defined as the argument where this shifted profile equals one half

Figure 2.  The panel on the left represents a graph $ j\mapsto \Gamma_j(t) $ at a fixed time $ t $. The right panel zooms in on three nodes of this graph to illustrate the identities (1.28) and (1.29) that underpin the drift term in our discrete curvature flow

Figure 3.  Both panels illustrate front-like initial conditions that satisfy (1.4) and hence fall within the framework of this paper. Panel a) provides an example of an initial perturbation that converges uniformly to a traveling front. On the contrary, the initial perturbation in b) does not uniformly converge to a traveling planar front, but the evolution of the interface is described asymptotically by (1.33)