# American Institute of Mathematical Sciences

• Previous Article
Long time dynamics of solutions to $p$-Laplacian diffusion problems with bistable reaction terms
• DCDS Home
• This Issue
• Next Article
On some model problem for the propagation of interacting species in a special environment
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  July 2021 Early access  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
##### References:

show all references

##### References:
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)
 [1] Matthieu Alfaro, Jérôme Coville, Gaël Raoul. Bistable travelling waves for nonlocal reaction diffusion equations. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1775-1791. doi: 10.3934/dcds.2014.34.1775 [2] Masaharu Taniguchi. Instability of planar traveling waves in bistable reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 21-44. doi: 10.3934/dcdsb.2003.3.21 [3] C. van der Mee, Stella Vernier Piro. Travelling waves for solid-gas reaction-diffusion systems. Conference Publications, 2003, 2003 (Special) : 872-879. doi: 10.3934/proc.2003.2003.872 [4] H. J. Hupkes, L. Morelli. Travelling corners for spatially discrete reaction-diffusion systems. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1609-1667. doi: 10.3934/cpaa.2020058 [5] Sheng-Chen Fu, Je-Chiang Tsai. Stability of travelling waves of a reaction-diffusion system for the acidic nitrate-ferroin reaction. Discrete & Continuous Dynamical Systems, 2013, 33 (9) : 4041-4069. doi: 10.3934/dcds.2013.33.4041 [6] Yuzo Hosono. Phase plane analysis of travelling waves for higher order autocatalytic reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 115-125. doi: 10.3934/dcdsb.2007.8.115 [7] Narcisa Apreutesei, Vitaly Volpert. Reaction-diffusion waves with nonlinear boundary conditions. Networks & Heterogeneous Media, 2013, 8 (1) : 23-35. doi: 10.3934/nhm.2013.8.23 [8] Zhao-Xing Yang, Guo-Bao Zhang, Ge Tian, Zhaosheng Feng. Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 581-603. doi: 10.3934/dcdss.2017029 [9] Sheng-Chen Fu. Travelling waves of a reaction-diffusion model for the acidic nitrate-ferroin reaction. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 189-196. doi: 10.3934/dcdsb.2011.16.189 [10] Xiongxiong Bao, Wan-Tong Li. Existence and stability of generalized transition waves for time-dependent reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3621-3641. doi: 10.3934/dcdsb.2020249 [11] Yong Jung Kim, Wei-Ming Ni, Masaharu Taniguchi. Non-existence of localized travelling waves with non-zero speed in single reaction-diffusion equations. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3707-3718. doi: 10.3934/dcds.2013.33.3707 [12] Wei-Jie Sheng, Wan-Tong Li. Multidimensional stability of time-periodic planar traveling fronts in bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2681-2704. doi: 10.3934/dcds.2017115 [13] Ana Carpio, Gema Duro. Explosive behavior in spatially discrete reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 693-711. doi: 10.3934/dcdsb.2009.12.693 [14] Marie Henry, Danielle Hilhorst, Masayasu Mimura. A reaction-diffusion approximation to an area preserving mean curvature flow coupled with a bulk equation. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 125-154. doi: 10.3934/dcdss.2011.4.125 [15] Klemens Fellner, Wolfang Prager, Bao Q. Tang. The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks. Kinetic & Related Models, 2017, 10 (4) : 1055-1087. doi: 10.3934/krm.2017042 [16] Wei Wang, Wanbiao Ma. Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3213-3235. doi: 10.3934/dcdsb.2018242 [17] Judith R. Miller, Huihui Zeng. Stability of traveling waves for systems of nonlinear integral recursions in spatial population biology. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 895-925. doi: 10.3934/dcdsb.2011.16.895 [18] Alessandro Audrito. Bistable reaction equations with doubly nonlinear diffusion. Discrete & Continuous Dynamical Systems, 2019, 39 (6) : 2977-3015. doi: 10.3934/dcds.2019124 [19] Xiaoyan Zhang, Yuxiang Zhang. Spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2625-2640. doi: 10.3934/dcdsb.2018124 [20] Henri Berestycki, Nancy Rodríguez. A non-local bistable reaction-diffusion equation with a gap. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 685-723. doi: 10.3934/dcds.2017029

2020 Impact Factor: 1.392

## Metrics

• HTML views (177)
• Cited by (0)

• on AIMS