# American Institute of Mathematical Sciences

June  2022, 21(6): 2189-2251. doi: 10.3934/cpaa.2022055

## Curvature-driven front propagation through planar lattices in oblique directions

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

*Corresponding author

Received  July 2021 Revised  January 2022 Published  June 2022 Early access  March 2022

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

In this paper we investigate the long-term behaviour of solutions to the discrete Allen-Cahn equation posed on a two-dimensional lattice. We show that front-like initial conditions evolve towards a planar travelling wave modulated by a phaseshift $\gamma_l(t)$ that depends on the coordinate $l$ transverse to the primary direction of propagation. This direction is allowed to be general, but rational, generalizing earlier known results for the horizontal direction. We show that the behaviour of $\gamma$ can be asymptotically linked to the behaviour of a suitably discretized mean curvature flow. This allows us to show that travelling waves propagating in rational directions are nonlinearly stable with respect to perturbations that are asymptotically periodic in the transverse direction.

Citation: Mia Jukić, Hermen Jan Hupkes. Curvature-driven front propagation through planar lattices in oblique directions. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2189-2251. doi: 10.3934/cpaa.2022055
##### References:
 [1] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5. [2] P. W. Bates and A. Chmaj, A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-368.  doi: 10.1007/s002050050189. [3] J. Bell, Some threshold results for models of myelinated nerves, Math. Bio., 54 (1981), 181-190.  doi: 10.1016/0025-5564(81)90085-7. [4] J. Bell and C. Cosner, Threshold behavior and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons, Quart. Appl. Math., 42 (1984), 1-14.  doi: 10.1090/qam/736501. [5] H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, Contemp. Math., 446 (2007), 101-124. [6] H. Berestycki, F. Hamel and H. Matano, Bistable traveling waves around an obstacle, Commun. Pure Appl. Math., 62 (2009), 729-788.  doi: 10.1002/cpa.20275. [7] J. W. Cahn, Theory of crystal growth and interface motion in crystalline materials, Acta met., 8 (1960), 554-562. [8] J. W. Cahn, J. Mallet-Paret and E. S. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Appl. Math., 59 (1999), 455-493.  doi: 10.1137/S0036139996312703. [9] R. Cerf, The Wulff Crystal in Ising and Percolation Models: Ecole D'Eté de Probabilités de Saint-Flour XXXIV-2004, Springer, 2006. [10] S. H. Chang, P. C. Cosman and L. B. Milstein, Chernoff-type bounds for the gaussian error function, IEEE Trans. Commun., 59 (2011), 2939-2944. [11] M. Chiani and D. Dardari, Improved exponential bounds and approximation for the q-function with application to average error probability computation, in Global Telecommunications Conference, IEEE, 2002. [12] H. Cook, D. D. Fontaine and J. E. Hilliard, A model for diffusion on cubic lattices and its application to the early stages of ordering, Acta Met., 17 (1969), 765-773. [13] P. Diaconis and L. Saloff-Coste, Convolution powers of complex functions on, Math. Nachr., 287 (2014), 1106-1130.  doi: 10.1002/mana.201200163. [14] P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer Science & Business Media, 2013. [15] P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361.  doi: 10.1007/BF00250432. [16] J.-S. Guo and F. Hamel, Front propagation for discrete periodic monostable equations, Math. Ann., 335 (2006), 489-525.  doi: 10.1007/s00208-005-0729-0. [17] M. Haragus and A. Scheel, Almost Planar Waves in Anisotropic Media, Commun. Partial Differ. Equ., 31 (2006), 791-815.  doi: 10.1080/03605300500361420. [18] A. Hoffman, H. Hupkes and E. Van Vleck, Multi-dimensional stability of waves travelling through rectangular lattices in rational directions, Trans. Amer. Math. Soc., 367 (2015), 8757-8808.  doi: 10.1090/S0002-9947-2015-06392-2. [19] A. Hoffman, H. Hupkes and E. Van Vleck, Entire Solutions for Bistable Lattice Differential Equations with Obstacles, American Mathematical Society, 2017. doi: 10.1090/memo/1188. [20] A. Hoffman and J. Mallet-Paret, Universality of crystallographic pinning, J. Dynam. Differ. Equ., 22 (2010), 79-119.  doi: 10.1007/s10884-010-9157-2. [21] H. J. Hupkes and B. Sandstede, Stability of pulse solutions for the discrete FitzHugh-Nagumo system, Trans. Amer. Math. Soc., 365 (2013), 251-301.  doi: 10.1090/S0002-9947-2012-05567-X. [22] H. J. Hupkes and L. Morelli, Travelling corners for spatially discrete reaction-diffusion system, Commun. Pure Appl. Anal., 19 (2020), 1609-1667. [23] H. J. Hupkes, L. Morelli, W. M. Schouten-Straatman and E. S. Van Vleck, Traveling waves and pattern formation for spatially discrete bistable reaction-diffusion equations, in International Conference on Difference Equations and Applications, Springer, 2018. [24] C. K. Jones, Spherically symmetric solutions of a reaction-diffusion equation, J. Differ. Equ., 49 (1983), 142-169.  doi: 10.1016/0022-0396(83)90023-2. [25] M. Jukić and H. J. Hupkes, Dynamics of curved travelling fronts for the discrete allen-cahn equation on a two-dimensional lattice, Discret. Contin. Dynam. Syst. A, 3163–3209. doi: 10.3934/dcds.2020402. [26] T. Kapitula, Multidimensional stability of planar travelling waves, Trans. Amer. Math. Soc., 349 (1997), 257-269.  doi: 10.1090/S0002-9947-97-01668-1. [27] J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572.  doi: 10.1137/0147038. [28] J. P. Keener and J. Sneyd, Mathematical Physiology, Springer, 1998. [29] T. H. Keitt, M. A. Lewis and R. D. Holt, Allee effects, invasion pinning, and species' borders, Amer. Nat., 157 (2001), 203-216. [30] P. G. Kevrekidis, Non-linear waves in lattices: past, present, future, IMA J. Appl. Math., 76 (2011), 389-423.  doi: 10.1093/imamat/hxr015. [31] C. D. Levermore and J. X. Xin, Multidimensional Stability of Travelling Waves in a Bistable Reaction-Diffusion Equation, Ⅱ, Commun. Partial Differ. Equ., 17 (1992), 1901-1924.  doi: 10.1080/03605309208820908. [32] J. Mallet-Paret, Crystallographic Pinning: Direction Dependent Pinning in Lattice Differential Equations, Citeseer, 2001. [33] J. Mallet-Paret, The fredholm alternative for functional differential equations of mixed type, Journal of Dynamics and Differential Equations, 11 (1999), 1-47.  doi: 10.1023/A:1021889401235. [34] J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems, Journal of Dynamics and Differential Equations, 11 (1999), 49-127.  doi: 10.1023/A:1021841618074. [35] H. Matano, Y. Mori and M. Nara, Asymptotic behavior of spreading fronts in the anisotropic allen–cahn equation on rn, Ann. I. H. Poincaré-An, 36 (2019), 585-626.  doi: 10.1016/j.anihpc.2018.07.003. [36] H. Matano and M. Nara, Large time behavior of disturbed planar fronts in the Allen–Cahn equation, J. Differ. Equ., 251 (2011), 3522-3557.  doi: 10.1016/j.jde.2011.08.029. [37] R. M. Merks, Y. Van de Peer, D. Inzé and G. T. Beemster, Canalization without flux sensors: a traveling-wave hypothesis, Trends Plant Sci., 12 (2007), 384-390. [38] S. Osher and B. Merriman, The wulff shape as the asymptotic limit of a growing crystalline interface, Asian J. Math., 1 (1997), 560-571.  doi: 10.4310/AJM.1997.v1.n3.a6. [39] E. Randles and L. Saloff-Coste, On the convolution powers of complex functions on $\mathbb{Z}$, J. Fourier Anal. Appl., 21 (2015), 754-798.  doi: 10.1007/s00041-015-9386-1. [40] V. Roussier, Stability of radially symmetric travelling waves in reaction–diffusion equations, Ann. I. H. Poincare (C)-An, 21 (2004), 341-379.  doi: 10.1016/S0294-1449(03)00042-8. [41] D. Sattinger, Weighted norms for the stability of traveling waves, J. Differ. Equ., 25 (1977), 130-144.  doi: 10.1016/0022-0396(77)90185-1. [42] C. M. Taylor and A. Hastings, Allee effects in biological invasions, Eco. Lett., 8 (2005), 895-908. [43] K. Uchiyama, Asymptotic behavior of solutions of reaction-diffusion equations with varying drift coefficients, Arch. Ration. Mech. Anal., 90 (1985), 291-311.  doi: 10.1007/BF00276293. [44] B. van Hal, Travelling Waves in Discrete Spatial Domains, Bachelor Thesis. [45] G. Wul, Achen on the question of the speed of growth and dissolution of the crystal, Z. Crystallogr, 34 (1901), 449-530. [46] J. X. Xin, Multidimensional Stability of Travelling Waves in a Bistable Reaction-Diffusion Equation, Ⅰ, Commun. Partial Differ. Equ., 17 (1992), 1889-1899.  doi: 10.1080/03605309208820907.

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##### References:
 [1] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5. [2] P. W. Bates and A. Chmaj, A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-368.  doi: 10.1007/s002050050189. [3] J. Bell, Some threshold results for models of myelinated nerves, Math. Bio., 54 (1981), 181-190.  doi: 10.1016/0025-5564(81)90085-7. [4] J. Bell and C. Cosner, Threshold behavior and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons, Quart. Appl. Math., 42 (1984), 1-14.  doi: 10.1090/qam/736501. [5] H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, Contemp. Math., 446 (2007), 101-124. [6] H. Berestycki, F. Hamel and H. Matano, Bistable traveling waves around an obstacle, Commun. Pure Appl. Math., 62 (2009), 729-788.  doi: 10.1002/cpa.20275. [7] J. W. Cahn, Theory of crystal growth and interface motion in crystalline materials, Acta met., 8 (1960), 554-562. [8] J. W. Cahn, J. Mallet-Paret and E. S. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Appl. Math., 59 (1999), 455-493.  doi: 10.1137/S0036139996312703. [9] R. Cerf, The Wulff Crystal in Ising and Percolation Models: Ecole D'Eté de Probabilités de Saint-Flour XXXIV-2004, Springer, 2006. [10] S. H. Chang, P. C. Cosman and L. B. Milstein, Chernoff-type bounds for the gaussian error function, IEEE Trans. Commun., 59 (2011), 2939-2944. [11] M. Chiani and D. Dardari, Improved exponential bounds and approximation for the q-function with application to average error probability computation, in Global Telecommunications Conference, IEEE, 2002. [12] H. Cook, D. D. Fontaine and J. E. Hilliard, A model for diffusion on cubic lattices and its application to the early stages of ordering, Acta Met., 17 (1969), 765-773. [13] P. Diaconis and L. Saloff-Coste, Convolution powers of complex functions on, Math. Nachr., 287 (2014), 1106-1130.  doi: 10.1002/mana.201200163. [14] P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Springer Science & Business Media, 2013. [15] P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Ration. Mech. Anal., 65 (1977), 335-361.  doi: 10.1007/BF00250432. [16] J.-S. Guo and F. Hamel, Front propagation for discrete periodic monostable equations, Math. Ann., 335 (2006), 489-525.  doi: 10.1007/s00208-005-0729-0. [17] M. Haragus and A. Scheel, Almost Planar Waves in Anisotropic Media, Commun. Partial Differ. Equ., 31 (2006), 791-815.  doi: 10.1080/03605300500361420. [18] A. Hoffman, H. Hupkes and E. Van Vleck, Multi-dimensional stability of waves travelling through rectangular lattices in rational directions, Trans. Amer. Math. Soc., 367 (2015), 8757-8808.  doi: 10.1090/S0002-9947-2015-06392-2. [19] A. Hoffman, H. Hupkes and E. Van Vleck, Entire Solutions for Bistable Lattice Differential Equations with Obstacles, American Mathematical Society, 2017. doi: 10.1090/memo/1188. [20] A. Hoffman and J. Mallet-Paret, Universality of crystallographic pinning, J. Dynam. Differ. Equ., 22 (2010), 79-119.  doi: 10.1007/s10884-010-9157-2. [21] H. J. Hupkes and B. Sandstede, Stability of pulse solutions for the discrete FitzHugh-Nagumo system, Trans. Amer. Math. Soc., 365 (2013), 251-301.  doi: 10.1090/S0002-9947-2012-05567-X. [22] H. J. Hupkes and L. Morelli, Travelling corners for spatially discrete reaction-diffusion system, Commun. Pure Appl. Anal., 19 (2020), 1609-1667. [23] H. J. Hupkes, L. Morelli, W. M. Schouten-Straatman and E. S. Van Vleck, Traveling waves and pattern formation for spatially discrete bistable reaction-diffusion equations, in International Conference on Difference Equations and Applications, Springer, 2018. [24] C. K. Jones, Spherically symmetric solutions of a reaction-diffusion equation, J. Differ. Equ., 49 (1983), 142-169.  doi: 10.1016/0022-0396(83)90023-2. [25] M. Jukić and H. J. Hupkes, Dynamics of curved travelling fronts for the discrete allen-cahn equation on a two-dimensional lattice, Discret. Contin. Dynam. Syst. A, 3163–3209. doi: 10.3934/dcds.2020402. [26] T. Kapitula, Multidimensional stability of planar travelling waves, Trans. Amer. Math. Soc., 349 (1997), 257-269.  doi: 10.1090/S0002-9947-97-01668-1. [27] J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572.  doi: 10.1137/0147038. [28] J. P. Keener and J. Sneyd, Mathematical Physiology, Springer, 1998. [29] T. H. Keitt, M. A. Lewis and R. D. Holt, Allee effects, invasion pinning, and species' borders, Amer. Nat., 157 (2001), 203-216. [30] P. G. Kevrekidis, Non-linear waves in lattices: past, present, future, IMA J. Appl. Math., 76 (2011), 389-423.  doi: 10.1093/imamat/hxr015. [31] C. D. Levermore and J. X. Xin, Multidimensional Stability of Travelling Waves in a Bistable Reaction-Diffusion Equation, Ⅱ, Commun. Partial Differ. Equ., 17 (1992), 1901-1924.  doi: 10.1080/03605309208820908. [32] J. Mallet-Paret, Crystallographic Pinning: Direction Dependent Pinning in Lattice Differential Equations, Citeseer, 2001. [33] J. Mallet-Paret, The fredholm alternative for functional differential equations of mixed type, Journal of Dynamics and Differential Equations, 11 (1999), 1-47.  doi: 10.1023/A:1021889401235. [34] J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems, Journal of Dynamics and Differential Equations, 11 (1999), 49-127.  doi: 10.1023/A:1021841618074. [35] H. Matano, Y. Mori and M. Nara, Asymptotic behavior of spreading fronts in the anisotropic allen–cahn equation on rn, Ann. I. H. Poincaré-An, 36 (2019), 585-626.  doi: 10.1016/j.anihpc.2018.07.003. [36] H. Matano and M. Nara, Large time behavior of disturbed planar fronts in the Allen–Cahn equation, J. Differ. Equ., 251 (2011), 3522-3557.  doi: 10.1016/j.jde.2011.08.029. [37] R. M. Merks, Y. Van de Peer, D. Inzé and G. T. Beemster, Canalization without flux sensors: a traveling-wave hypothesis, Trends Plant Sci., 12 (2007), 384-390. [38] S. Osher and B. Merriman, The wulff shape as the asymptotic limit of a growing crystalline interface, Asian J. Math., 1 (1997), 560-571.  doi: 10.4310/AJM.1997.v1.n3.a6. [39] E. Randles and L. Saloff-Coste, On the convolution powers of complex functions on $\mathbb{Z}$, J. Fourier Anal. Appl., 21 (2015), 754-798.  doi: 10.1007/s00041-015-9386-1. [40] V. Roussier, Stability of radially symmetric travelling waves in reaction–diffusion equations, Ann. I. H. Poincare (C)-An, 21 (2004), 341-379.  doi: 10.1016/S0294-1449(03)00042-8. [41] D. Sattinger, Weighted norms for the stability of traveling waves, J. Differ. Equ., 25 (1977), 130-144.  doi: 10.1016/0022-0396(77)90185-1. [42] C. M. Taylor and A. Hastings, Allee effects in biological invasions, Eco. Lett., 8 (2005), 895-908. [43] K. Uchiyama, Asymptotic behavior of solutions of reaction-diffusion equations with varying drift coefficients, Arch. Ration. Mech. Anal., 90 (1985), 291-311.  doi: 10.1007/BF00276293. [44] B. van Hal, Travelling Waves in Discrete Spatial Domains, Bachelor Thesis. [45] G. Wul, Achen on the question of the speed of growth and dissolution of the crystal, Z. Crystallogr, 34 (1901), 449-530. [46] J. X. Xin, Multidimensional Stability of Travelling Waves in a Bistable Reaction-Diffusion Equation, Ⅰ, Commun. Partial Differ. Equ., 17 (1992), 1889-1899.  doi: 10.1080/03605309208820907.
Both panels show the sublattice ${\mathbb{Z}^2_{\times}}$ obtained after the coordinate transformation (1.3), for the rational direction $(\sigma_h, \sigma_v) = (2, 3)$ on the left and the irrational angle $\pi/6$ on the right. We see that the left lattice is a proper subset of $\mathbb{Z}^2$. On the right however the purple dots only coincide with $\mathbb{Z}^2$ at the origin. Moreover, the sets $\{i\sigma_h + j\sigma_v: (i,j)\in \mathbb{Z}^2\}$ and $\{i\sigma_v - j\sigma_h: (i,j)\in \mathbb{Z}^2\}$ are both dense in ${\mathbb R}$. This feature significantly differentiates the analysis between the rational and irrational directions
Here we provide the geometric motivation behind the definition (1.23) for $\overline{c}_\Gamma$ with $N = 2$. Since there is no uniquely defined normal direction for discrete graphs, we take the average of the velocities associated to the directions transverse to the connecting lines between $(\Gamma_l,l)$ and $(\Gamma_{l+k} , l+k)$. Here we consider each $0 < |k| \le 2$
Each row represents a time evolution of the solution $u$ to the Allen-Cahn equation (1.1) with the cubic nonlinearity (1.2). Both initial conditions satisfy (H$0$), but the bottom row also satisfies the assumptions of Theorem 2.10. Proposition 2.6 states that $u$ converges to a wavefront travelling in the $n$-direction with a phase $\gamma_l(t)$, which in the bottom row becomes homogeneous with respect to $l$ on account of (2.34)
In order to construct the phase $\gamma_l(t)$ for a fixed pair $(l,t)$ we first identify an interfacial region around the value $\Phi_*(0) = \frac{1}{2}$ (shaded in blue) where the (discrete) function $n \mapsto u_{n,l}(t)$ is monotone. We subsequently stretch the waveprofile to match the (pink) points $\left(n_*, u_{n_*, l}(t)\right)$ and $\left(n_*+\sigma_*^2, u_{n_*+\sigma_*^2, l}(t)\right)$ introduced in (2.24)
These plots represent the outcome of our numerical computations for the values $M_{(\sigma_h, \sigma_v)}$, where we used $g(u;a) = 6 u(u-1)(u-a)$ with $a = 0.45$. For each fixed $\sigma_h$ (horizontal) we computed these values for each integer $1 \le \sigma_v \le \sigma_h$ that has $\mathrm{gcd}(\sigma_h, \sigma_v) = 1$, using the color code to represent the fraction ${\sigma_v}/{\sigma_h}$. On the left we see the formation of horizontal bands of the same color, suggesting the possibility to take limits along convergent subsequences $({\sigma_v^{(n)}}/{\sigma_h^{(n)}})_{n > 0}$; see also Figure 6. The $\sigma_*^2$-scaling on the right shows that our condition requiring $M_{(\sigma_h, \sigma_v)}$ to be negative can be confirmed in a robust fashion
These plots track the values of $M_{(\sigma_h, \sigma_v)}$ along several subsequences of fractions $\sigma_v / \sigma_h$ that converge to $4/5$ (left) or zero (right). In all cases the limits are strictly above the values $M_{(5,4)}$ (left) and $M_{(1,0)}$ (right) corresponding to the limiting angles, supporting the inequality (2.36)
These six graphs represent the time evolution of the Green's function $M_l(t)$, which we computed numerically by applying (5.16) to the coefficients $(a_k)_{k = -10}^{10}$ appearing in Table 1. Observe the negative values for $M_l(t)$ that are clearly visible for $t = 0.1$, together with the leftward movement of the 'center of mass', which travels at the speed $-a\cdot\mu = -0.22$
Numerically computed values for the coefficients $(a_k)$ defined in (2.19) for the propagation direction $(\sigma_h, \sigma_v) = (2,5)$, with the nonlinearity $g(u;a) = 6 u(u-1)(u-0.45)$. We computed these coefficients for a large range of angles and used them to calculate the values $M_{(\sigma_h, \sigma_v)}$ depicted in Figures 5-6
 k 1 2 3 4 5 6 7 8 9 10 ak 0 0.896 0.195 0.068 0.966 0 -0.143 0 0 0.05 a-k 0 0.912 0.0925 0.179 1.005 0 -0.199 0 0 0.03
 k 1 2 3 4 5 6 7 8 9 10 ak 0 0.896 0.195 0.068 0.966 0 -0.143 0 0 0.05 a-k 0 0.912 0.0925 0.179 1.005 0 -0.199 0 0 0.03
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