Advanced Search
Article Contents
Article Contents

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

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

    * Corresponding author: m.jukic@math.leidenuniv.nl 
Both authors acknowledge support from the Netherlands Organization for Scientific Research (NWO) (grant 639.032.612)
Abstract Full Text(HTML) Figure(3) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: 34K31, 37L15.


    \begin{equation} \\ \end{equation}
  • 加载中
  • 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)

  • [1] S. M. Allen and J. W. Cahn, A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta Metallurgica, 27 (1979), 1085-1095.  doi: 10.1016/0001-6160(79)90196-2.
    [2] D. E. Amos, Computation of modified bessel functions and their ratios, Mathematics of Computation, 28 (1974), 239-251.  doi: 10.1090/S0025-5718-1974-0333287-7.
    [3] D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in Partial Differential Equations and Related Topics, Springer, 466 (1975), 5–49.
    [4] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Advances in Mathematics, 30 (1978), 33-76.  doi: 10.1016/0001-8708(78)90130-5.
    [5] M. Bär, M. Falcke, H. Levine and L. S. Tsimring, Discrete stochastic modeling of calcium channel dynamics, Physical Review Letters, 84 (2000), 5664.
    [6] I. Barashenkov, O. Oxtoby and D. E. Pelinovsky, Translationally invariant discrete kinks from one-dimensional maps, Physical Review E, 72 (2005), 035602, 4pp. doi: 10.1103/PhysRevE.72.035602.
    [7] P. W. Bates and A. Chmaj, A discrete convolution model for phase transitions, Archive for Rational Mechanics and Analysis, 150 (1999), 281-305.  doi: 10.1007/s002050050189.
    [8] M. BeckB. Sandstede and K. Zumbrun, Nonlinear stability of time-periodic viscous shocks, Archive for Rational Mechanics and Analysis, 196 (2010), 1011-1076.  doi: 10.1007/s00205-009-0274-1.
    [9] J. Bell, Some threshold results for models of myelinated nerves, Mathematical Biosciences, 54 (1981), 181-190.  doi: 10.1016/0025-5564(81)90085-7.
    [10] J. Bell and C. Cosner, Threshold behavior and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons, Quarterly of Applied Mathematics, 42 (1984), 1-14.  doi: 10.1090/qam/736501.
    [11] H. Berestycki and F. Hamel, Generalized travelling waves for reaction-diffusion equations, Contemporary Mathematics, 446 (2007), 101-123.  doi: 10.1090/conm/446/08627.
    [12] H. BerestyckiF. Hamel and H. Matano, Bistable traveling waves around an obstacle, Comm. Pure Appl. Math., 62 (2009), 729-788.  doi: 10.1002/cpa.20275.
    [13] H. BerestyckiH. Matano and F. Hamel, Bistable traveling waves around an obstacle, Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, 62 (2009), 729-788.  doi: 10.1002/cpa.20275.
    [14] J. W. Cahn, Theory of crystal growth and interface motion in crystalline materials, Acta metallurgica, 8 (1960), 554-562. 
    [15] A. Carpio, L. Bonilla and G. Dell'Acqua, Motion of wave fronts in semiconductor superlattices, Physical Review E, 64 (2001), 036204. doi: 10.1103/PhysRevE.64.036204.
    [16] X. ChenJ. S. Guo and C. C. Wu, Traveling waves in discrete periodic media for bistable dynamics, Arch. Ration. Mech. Anal., 189 (2008), 189-236.  doi: 10.1007/s00205-007-0103-3.
    [17] S.-N. Chow and J. Mallet-Paret, Pattern formation and spatial chaos in lattice dynamical systems. Ⅰ, IEEE Transactions on Circuits and Systems Ⅰ: Fundamental Theory and Applications, 42 (1995), 746-751.  doi: 10.1109/81.473583.
    [18] S.-N. ChowJ. Mallet-Paret and W. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291.  doi: 10.1006/jdeq.1998.3478.
    [19] S.-N. ChowJ. Mallet-Paret and E. S. Van Vleck, Dynamics of lattice differential equations, International Journal of Bifurcation and Chaos, 6 (1996), 1605-1621.  doi: 10.1142/S0218127496000977.
    [20] K. Crane, Discrete differential geometry: An applied introduction, Notices of the AMS, Communication, 1153–1159.
    [21] K. DeckelnickG. Dziuk and C. M. Elliott, Computation of geometric partial differential equations and mean curvature flow, Acta numerica, 14 (2005), 139-232.  doi: 10.1017/S0962492904000224.
    [22] W. Ding and T. Giletti, Admissible speeds in spatially periodic bistable reaction-diffusion equations, arXiv preprint, arXiv: 2006.05118.
    [23] S. V. DmitrievP. G. Kevrekidis and N. Yoshikawa, Discrete Klein–Gordon models with static kinks free of the Peierls–Nabarro potential, J. Phys. A., 38 (2005), 7617-7627.  doi: 10.1088/0305-4470/38/35/002.
    [24] P. C. Fife, Long time behavior of solutions of bistable nonlinear diffusion equationsn, Archive for Rational Mechanics and Analysis, 70 (1979), 31-46.  doi: 10.1007/BF00276380.
    [25] P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, 28. Springer-Verlag, Berlin-New York, 1979.
    [26] P. C. Fife and J. B. McLeod, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Archive for Rational Mechanics and Analysis, 65 (1977), 335-361.  doi: 10.1007/BF00250432.
    [27] T. Gallay, E. Risler et al., A variational proof of global stability for bistable travelling waves, Differential and integral equations, 20 (2007), 901-926.
    [28] G. N. Watson, A Treatise On The Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944.
    [29] D. Hankerson and B. Zinner, Wavefronts for a cooperative tridiagonal system of differential equations, Journal of Dynamics and Differential Equations, 5 (1993), 359-373.  doi: 10.1007/BF01053165.
    [30] A. HoffmanH. J. Hupkes and E. S. Van Vleck, Multi-dimensional stability of waves travelling through rectangular lattices in rational directions, Transactions of the American Mathematical Society, 367 (2015), 8757-8808.  doi: 10.1090/S0002-9947-2015-06392-2.
    [31] A. Hoffman, H. J. Hupkes and E. S. Van Vleck, Entire Solutions for Bistable Lattice Differential Equations with Obstacles, vol. 250, American Mathematical Society, 2017. doi: 10.1090/memo/1188.
    [32] A. Hoffman and J. Mallet-Paret, Universality of crystallographic pinning, J. Dynam. Differential Equations, 22 (2010), 79-119.  doi: 10.1007/s10884-010-9157-2.
    [33] H. J. Hupkes and L. Morelli, Travelling corners for spatially discrete reaction-diffusion systems, Communications on Pure and Applied Analysis, 19 (2020), 1609-1667.  doi: 10.3934/cpaa.2020058.
    [34] H. J. HupkesD. Pelinovsky and B. Sandstede, Propagation failure in the discrete Nagumo equation, Proc. Amer. Math. Soc., 139 (2011), 3537-3551.  doi: 10.1090/S0002-9939-2011-10757-3.
    [35] 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.
    [36] C. K. Jones, Spherically symmetric solutions of a reaction-diffusion equation, Journal of Differential Equations, 49 (1983), 142-169.  doi: 10.1016/0022-0396(83)90023-2.
    [37] T. Kapitula, Multidimensional stability of planar travelling waves, Transactions of the American Mathematical Society, 349 (1997), 257-269.  doi: 10.1090/S0002-9947-97-01668-1.
    [38] J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM Journal on Applied Mathematics, 47 (1987), 556-572.  doi: 10.1137/0147038.
    [39] J. P. Keener, The effects of discrete gap junction coupling on propagation in myocardium, Journal of Theoretical Biology, 148 (1991), 49-82.  doi: 10.1016/S0022-5193(05)80465-5.
    [40] T. H. KeittM. A. Lewis and R. D. Holt, Allee effects, invasion pinning, and species' borders, The American Naturalist, 157 (2001), 203-216.  doi: 10.1086/318633.
    [41] C. D. Levermore and J. X. Xin, Multidimensional stability of travelling waves in a bistable reaction-diffusion equation, Ⅱ, Comm. PDE, 17 (1992), 1901-1924.  doi: 10.1080/03605309208820908.
    [42] S. A. Levin, Population dynamic models in heterogeneous environments, Annual review of ecology and systematics, 7 (1976), 287-310.  doi: 10.1146/annurev.es.07.110176.001443.
    [43] J. Mallet-Paret, Crystallographic Pinning: Direction Dependent Pinning in Lattice Differential Equations, Citeseer, 2001.
    [44] 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.
    [45] 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.
    [46] H. Matano, Y. Mori and M. Nara, Asymptotic behavior of spreading fronts in the anisotropic allen–cahn equation on rn,, in Annales de l'Institut Henri Poincaré C, Analyse non Linéaire, Elsevier, 36 (2019), 585–626. doi: 10.1016/j.anihpc.2018.07.003.
    [47] H. Matano and M. Nara, Large time behavior of disturbed planar fronts in the allen–cahn equation, Journal of Differential Equations, 251 (2011), 3522-3557.  doi: 10.1016/j.jde.2011.08.029.
    [48] E. Neuman, Inequalities involving modified bessel functions of the first kind, Journal of Mathematical Analysis and Applications, 171 (1992), 532-536.  doi: 10.1016/0022-247X(92)90363-I.
    [49] B. V. Pal'tsev, Two-sided bounds uniform in the real argument and the index for modified bessel functions, Mathematical Notes, 65 (1999), 571-581.  doi: 10.1007/BF02743167.
    [50] E. Risler, Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure,, in Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol. 25, Elsevier, 2008,381–424. doi: 10.1016/j.anihpc.2006.12.005.
    [51] V. Roussier, Stability of radially symmetric travelling waves in reaction–diffusion equations,, in Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol. 21, Elsevier, 2004,341–379. doi: 10.1016/S0294-1449(03)00042-8.
    [52] W. M. Schouten-Straatman and H. J. Hupkes, Nonlinear stability of pulse solutions for the discrete Fitzhugh-Nagumo equation with infinite-range interactions, Discrete and Continuous Dynamical Systems A, 39 (2019), 5017-5083.  doi: 10.3934/dcds.2019205.
    [53] R. P. Soni, On an inequality for modified bessel functions, Journal of Mathematics and Physics, 44 (1965), 406-407.  doi: 10.1002/sapm1965441406.
    [54] G.-Q. Sun, Mathematical modeling of population dynamics with allee effect, Nonlinear Dynamics, 85 (2016), 1-12.  doi: 10.1007/s11071-016-2671-y.
    [55] C. M. Taylor and A. Hastings, Allee effects in biological invasions, Ecology Letters, 8 (2005), 895-908.  doi: 10.1111/j.1461-0248.2005.00787.x.
    [56] K. Uchiyama, Asymptotic behavior of solutions of reaction-diffusion equations with varying drift coefficients, Archive for Rational Mechanics and Analysis, 90 (1985), 291-311.  doi: 10.1007/BF00276293.
    [57] J. X. Xin, Multidimensional stability of travelling waves in a bistable reaction-diffusion equation, Ⅰ, Comm. PDE, 17 (1992), 1889-1899.  doi: 10.1080/03605309208820907.
    [58] H. Zeng, Stability of planar travelling waves for bistable reaction–diffusion equations in multiple dimensions, Applicable Analysis, 93 (2014), 653-664.  doi: 10.1080/00036811.2013.797075.
    [59] B. Zinner, Stability of traveling wavefronts for the discrete nagumo equation, SIAM journal on Mathematical Analysis, 22 (1991), 1016-1020.  doi: 10.1137/0522066.
    [60] B. Zinner, Existence of traveling wavefront solutions for the discrete nagumo equation, Journal of Differential Equations, 96 (1992), 1-27.  doi: 10.1016/0022-0396(92)90142-A.
  • 加载中



Article Metrics

HTML views(508) PDF downloads(237) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint