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July  2021, 41(7): 3163-3209. doi: 10.3934/dcds.2020402

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

 Mathematisch Instituut - Universiteit Leiden, P.O. Box 9512, 2300 RA Leiden; The Netherlands

* Corresponding author: m.jukic@math.leidenuniv.nl

Received  April 2020 Revised  October 2020 Published  December 2020

Fund Project: Both authors acknowledge support from the Netherlands Organization for Scientific Research (NWO) (grant 639.032.612)

In this paper we consider the discrete Allen-Cahn equation posed on a two-dimensional rectangular lattice. We analyze the large-time behaviour of solutions that start as bounded perturbations to the well-known planar front solution that travels in the horizontal direction. In particular, we construct an asymptotic phase function $\gamma_j(t)$ and show that for each vertical coordinate $j$ the corresponding horizontal slice of the solution converges to the planar front shifted by $\gamma_j(t)$. We exploit the comparison principle to show that the evolution of these phase variables can be approximated by an appropriate discretization of the mean curvature flow with a direction-dependent drift term. This generalizes the results obtained in [47] for the spatially continuous setting. Finally, we prove that the horizontal planar wave is nonlinearly stable with respect to perturbations that are asymptotically periodic in the vertical direction.

Citation: Mia Jukić, Hermen Jan Hupkes. Dynamics of curved travelling fronts for the discrete Allen-Cahn equation on a two-dimensional lattice. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3163-3209. doi: 10.3934/dcds.2020402
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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
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
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)
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