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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
References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

M. Bär, M. Falcke, H. Levine and L. S. Tsimring, Discrete stochastic modeling of calcium channel dynamics, Physical Review Letters, 84 (2000), 5664. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. W. Cahn, Theory of crystal growth and interface motion in crystalline materials, Acta metallurgica, 8 (1960), 554-562.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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K. Crane, Discrete differential geometry: An applied introduction, Notices of the AMS, Communication, 1153–1159. Google Scholar

[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.  Google Scholar

[22]

W. Ding and T. Giletti, Admissible speeds in spatially periodic bistable reaction-diffusion equations, arXiv preprint, arXiv: 2006.05118. Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[25]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, 28. Springer-Verlag, Berlin-New York, 1979.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

G. N. Watson, A Treatise On The Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

J. Mallet-Paret, Crystallographic Pinning: Direction Dependent Pinning in Lattice Differential Equations, Citeseer, 2001. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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G.-Q. Sun, Mathematical modeling of population dynamics with allee effect, Nonlinear Dynamics, 85 (2016), 1-12.  doi: 10.1007/s11071-016-2671-y.  Google Scholar

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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.  Google Scholar

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show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

M. Bär, M. Falcke, H. Levine and L. S. Tsimring, Discrete stochastic modeling of calcium channel dynamics, Physical Review Letters, 84 (2000), 5664. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. W. Cahn, Theory of crystal growth and interface motion in crystalline materials, Acta metallurgica, 8 (1960), 554-562.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

K. Crane, Discrete differential geometry: An applied introduction, Notices of the AMS, Communication, 1153–1159. Google Scholar

[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.  Google Scholar

[22]

W. Ding and T. Giletti, Admissible speeds in spatially periodic bistable reaction-diffusion equations, arXiv preprint, arXiv: 2006.05118. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

P. C. Fife, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, 28. Springer-Verlag, Berlin-New York, 1979.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

G. N. Watson, A Treatise On The Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

J. Mallet-Paret, Crystallographic Pinning: Direction Dependent Pinning in Lattice Differential Equations, Citeseer, 2001. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[53]

R. P. Soni, On an inequality for modified bessel functions, Journal of Mathematics and Physics, 44 (1965), 406-407.  doi: 10.1002/sapm1965441406.  Google Scholar

[54]

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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)
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