American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Upper and lower estimates for invariance entropy
  • DCDS Home
  • This Issue
  • Next Article
    Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay
April  2011, 30(1): 137-167. doi: 10.3934/dcds.2011.30.137

Existence and uniqueness of traveling waves in a class of unidirectional lattice differential equations

Aaron Hoffman 1, and  Benjamin Kennedy 2,

1. 

Franklin W. Olin College of Engineering, 1000 Olin Way, Needham, MA 02492-1200, United States
2. 

Gettysburg College, Department of Mathematics, 300 North Washington St., Gettysburg, PA 17325-1400, United States

Received  October 2009 Revised  November 2010 Published  February 2011

We prove the existence and uniqueness, for wave speeds sufficiently large, of monotone traveling wave solutions connecting stable to unstable spatial equilibria for a class of $N$-dimensional lattice differential equations with unidirectional coupling. This class of lattice equations includes some cellular neural networks, monotone systems, and semi-discretizations for hyperbolic conservation laws with a source term. We obtain a variational characterization of the critical wave speed above which monotone traveling wave solutions are guaranteed to exist. We also discuss non-monotone waves, and the coexistence of monotone and non-monotone waves.
Keywords: monostable., traveling wave, Lattice differential equation.
Mathematics Subject Classification: Primary: 37L60, 34K10; Secondary: 34C19, 34K19.
Citation: Aaron Hoffman, Benjamin Kennedy. Existence and uniqueness of traveling waves in a class of unidirectional lattice differential equations. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 137-167. doi: 10.3934/dcds.2011.30.137
References:
[1]

P. Bates, X. Chen and A. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546. doi: 10.1137/S0036141000374002.

[2]

S. Benzoni-Gavage, Semi-discrete shock profiles for hyperbolic systems of conservation laws, Phys. D, 115 (1998), 109-123. doi: 10.1016/S0167-2789(97)00225-X.

[3]

S. Bianchini, BV solutions of semidiscrete upwind scheme, in "Hyperbolic problems: Theory, Numerics, Applications," Springer, (2003), 135-142.

[4]

J. W. Cahn, J. Mallet-Paret and E. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Appl. Math, 59 (1999), 455-493.

[5]

J. O. Chua and L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits and Systems, 35 (1998), 1257-1272. doi: 10.1109/31.7600.

[6]

J. O. Chua and L. Yang, Cellular neural networks: applications, IEEE Trans. Circuits and Systems, 35 (1998), 1273-1290. doi: 10.1109/31.7601.

[7]

J. L. Daleckii and M. G. Krein, "Stability of Solutions of Differential Equations in Banach Space,", Translated from the Russian by S.Smith, 43 (). 

[8]

O. Diekmann, S. A. van Gils, S. M. Verduyn-Lunel and H.-O. Walther, "Delay Equations: Functional- Complex- and Nonlinear Analysis," volume 110 of Applied Mathematical Sciences, Springer-Verlag, New York 1995.

[9]

U. Ebert, W. Van Saarloos and L. A. Peletier, Universal algebraic convergence in time of pulled fronts: the common mechanism for difference-differential and partial differential equations, European J. Appl. Math, 13 (2002), 53-66. doi: 10.1017/S0956792501004673.

[10]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 335-369.

[11]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," volume 99 of Applied Mathematical Sciences, Springer-Verlag, New York, 1993.

[12]

D. Henry, Small solutions of linear autonomous functional differential equations, J. Differential Equations, 8 (1970), 494-501. doi: 10.1016/0022-0396(70)90021-5.

[13]

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.

[14]

S.-S. Hsu and C.-H. Lin, Existence and multiplicity of traveling waves in a lattice dynamical system, J. Differential Equations, 164 (2000), 431-450. doi: 10.1006/jdeq.2000.3770.

[15]

S.-S. Hsu, C.-H. Lin and W. Shen, Traveling waves in cellular neural networks, International Journal of Bifurcation and Chaos, 9 (1999), 1307-1319. doi: 10.1142/S0218127499000912.

[16]

S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789. doi: 10.1137/070703016.

[17]

J. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572. doi: 10.1137/0147038.

[18]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l'equation de la diffusion avec croissance de la quantite de matiere et som application a un probleme biologique, Bull. Universite d'Etat a Moscou Ser. Int., Sect. A., 1 (1937), 1-25.

[19]

T. Krisztin, Global dynamics of delay differential equations, Period. Math. Hungar., 56 (2008), 83-95. doi: 10.1007/s10998-008-5083-x.

[20]

B. Li, M. A. Lewis and H. F. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions, J. Math. Biol., 58 (2009), 323-338. doi: 10.1007/s00285-008-0175-1.

[21]

J. Mallet-Paret, Morse decompositions for delay-differential equations, J. Differential Equations, 72 (1988), 270-315. doi: 10.1016/0022-0396(88)90157-X.

[22]

J. Mallet-Paret, The Fredholm alternative for functional-differential equations of mixed type, J. Dynam. Differential Equations, 11 (1999), 1-47. doi: 10.1023/A:1021889401235.

[23]

J. Mallet-Paret, Traveling waves in spatially discrete dynamical systems of diffusive type, in "Dynamical Systems," volume 1822 of Lecture Notes in Math., Springer, (2003), 231-298.

[24]

J. Mallet-Paret and G. Sell, The Poincare-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125 (1996), 441-489. doi: 10.1006/jdeq.1996.0037.

[25]

C. Mascia, Qualitative behavior of conservation laws with reaction term and nonconvex flux, Quart. Appl. Math., 58 (2000), 739-761.

[26]

R. D. Nussbaum, Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Mat. Pura Appl., 4 (1974), 263-306.

[27]

L. A. Peletier and J. A. Rodriguez, Fronts on a lattice, Differential Integral Equations, 17 (2004), 1013-1042.

[28]

W. Rudin, "Principles of Mathematical Analysis," McGraw-Hill, 1976.

[29]

D. Serre, Discrete shock profiles: Existence and stability, in "Hyperbolic Systems of Balance Laws," volume 1911 of Lecture Notes in Math., Springer, (2007), 79-158.

[30]

W. van Saarloos, Front Propagation into unstable states, Phys. Rep., 386 (2003), 29-222. doi: 10.1016/j.physrep.2003.08.001.

[31]

R. van Zon, H. van Beijeren and Ch. Dellago, Largest Lyapunov exponent for many particle systems at low densities, Phys. Rev. Lett., 80 (1998), 2035-2038. doi: 10.1103/PhysRevLett.80.2035.

[32]

H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. doi: 10.1137/0513028.

show all references

References:
[1]

P. Bates, X. Chen and A. Chmaj, Traveling waves of bistable dynamics on a lattice, SIAM J. Math. Anal., 35 (2003), 520-546. doi: 10.1137/S0036141000374002.

[2]

S. Benzoni-Gavage, Semi-discrete shock profiles for hyperbolic systems of conservation laws, Phys. D, 115 (1998), 109-123. doi: 10.1016/S0167-2789(97)00225-X.

[3]

S. Bianchini, BV solutions of semidiscrete upwind scheme, in "Hyperbolic problems: Theory, Numerics, Applications," Springer, (2003), 135-142.

[4]

J. W. Cahn, J. Mallet-Paret and E. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Appl. Math, 59 (1999), 455-493.

[5]

J. O. Chua and L. Yang, Cellular neural networks: Theory, IEEE Trans. Circuits and Systems, 35 (1998), 1257-1272. doi: 10.1109/31.7600.

[6]

J. O. Chua and L. Yang, Cellular neural networks: applications, IEEE Trans. Circuits and Systems, 35 (1998), 1273-1290. doi: 10.1109/31.7601.

[7]

J. L. Daleckii and M. G. Krein, "Stability of Solutions of Differential Equations in Banach Space,", Translated from the Russian by S.Smith, 43 (). 

[8]

O. Diekmann, S. A. van Gils, S. M. Verduyn-Lunel and H.-O. Walther, "Delay Equations: Functional- Complex- and Nonlinear Analysis," volume 110 of Applied Mathematical Sciences, Springer-Verlag, New York 1995.

[9]

U. Ebert, W. Van Saarloos and L. A. Peletier, Universal algebraic convergence in time of pulled fronts: the common mechanism for difference-differential and partial differential equations, European J. Appl. Math, 13 (2002), 53-66. doi: 10.1017/S0956792501004673.

[10]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 335-369.

[11]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional-Differential Equations," volume 99 of Applied Mathematical Sciences, Springer-Verlag, New York, 1993.

[12]

D. Henry, Small solutions of linear autonomous functional differential equations, J. Differential Equations, 8 (1970), 494-501. doi: 10.1016/0022-0396(70)90021-5.

[13]

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.

[14]

S.-S. Hsu and C.-H. Lin, Existence and multiplicity of traveling waves in a lattice dynamical system, J. Differential Equations, 164 (2000), 431-450. doi: 10.1006/jdeq.2000.3770.

[15]

S.-S. Hsu, C.-H. Lin and W. Shen, Traveling waves in cellular neural networks, International Journal of Bifurcation and Chaos, 9 (1999), 1307-1319. doi: 10.1142/S0218127499000912.

[16]

S.-B. Hsu and X.-Q. Zhao, Spreading speeds and traveling waves for nonmonotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789. doi: 10.1137/070703016.

[17]

J. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), 556-572. doi: 10.1137/0147038.

[18]

A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Etude de l'equation de la diffusion avec croissance de la quantite de matiere et som application a un probleme biologique, Bull. Universite d'Etat a Moscou Ser. Int., Sect. A., 1 (1937), 1-25.

[19]

T. Krisztin, Global dynamics of delay differential equations, Period. Math. Hungar., 56 (2008), 83-95. doi: 10.1007/s10998-008-5083-x.

[20]

B. Li, M. A. Lewis and H. F. Weinberger, Existence of traveling waves for integral recursions with nonmonotone growth functions, J. Math. Biol., 58 (2009), 323-338. doi: 10.1007/s00285-008-0175-1.

[21]

J. Mallet-Paret, Morse decompositions for delay-differential equations, J. Differential Equations, 72 (1988), 270-315. doi: 10.1016/0022-0396(88)90157-X.

[22]

J. Mallet-Paret, The Fredholm alternative for functional-differential equations of mixed type, J. Dynam. Differential Equations, 11 (1999), 1-47. doi: 10.1023/A:1021889401235.

[23]

J. Mallet-Paret, Traveling waves in spatially discrete dynamical systems of diffusive type, in "Dynamical Systems," volume 1822 of Lecture Notes in Math., Springer, (2003), 231-298.

[24]

J. Mallet-Paret and G. Sell, The Poincare-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125 (1996), 441-489. doi: 10.1006/jdeq.1996.0037.

[25]

C. Mascia, Qualitative behavior of conservation laws with reaction term and nonconvex flux, Quart. Appl. Math., 58 (2000), 739-761.

[26]

R. D. Nussbaum, Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Mat. Pura Appl., 4 (1974), 263-306.

[27]

L. A. Peletier and J. A. Rodriguez, Fronts on a lattice, Differential Integral Equations, 17 (2004), 1013-1042.

[28]

W. Rudin, "Principles of Mathematical Analysis," McGraw-Hill, 1976.

[29]

D. Serre, Discrete shock profiles: Existence and stability, in "Hyperbolic Systems of Balance Laws," volume 1911 of Lecture Notes in Math., Springer, (2007), 79-158.

[30]

W. van Saarloos, Front Propagation into unstable states, Phys. Rep., 386 (2003), 29-222. doi: 10.1016/j.physrep.2003.08.001.

[31]

R. van Zon, H. van Beijeren and Ch. Dellago, Largest Lyapunov exponent for many particle systems at low densities, Phys. Rev. Lett., 80 (1998), 2035-2038. doi: 10.1103/PhysRevLett.80.2035.

[32]

H. F. Weinberger, Long-time behavior of a class of biological models, SIAM J. Math. Anal., 13 (1982), 353-396. doi: 10.1137/0513028.

[1]

E. S. Van Vleck, Aijun Zhang. Competing interactions and traveling wave solutions in lattice differential equations. Communications on Pure and Applied Analysis, 2016, 15 (2) : 457-475. doi: 10.3934/cpaa.2016.15.457
[2]

Cheng-Hsiung Hsu, Ting-Hui Yang. Traveling plane wave solutions of delayed lattice differential systems in competitive Lotka-Volterra type. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 111-128. doi: 10.3934/dcdsb.2010.14.111
[3]

Rui Huang, Ming Mei, Yong Wang. Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3621-3649. doi: 10.3934/dcds.2012.32.3621
[4]

Cheng-Hsiung Hsu, Jian-Jhong Lin. Stability analysis of traveling wave solutions for lattice reaction-diffusion equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1757-1774. doi: 10.3934/dcdsb.2020001
[5]

Jong-Shenq Guo, Ying-Chih Lin. Traveling wave solution for a lattice dynamical system with convolution type nonlinearity. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 101-124. doi: 10.3934/dcds.2012.32.101
[6]

Lianzhang Bao, Zhengfang Zhou. Traveling wave solutions for a one dimensional model of cell-to-cell adhesion and diffusion with monostable reaction term. Discrete and Continuous Dynamical Systems - S, 2017, 10 (3) : 395-412. doi: 10.3934/dcdss.2017019
[7]

Tzong-Yow Lee and Fred Torcaso. Wave propagation in a lattice KPP equation in random media. Electronic Research Announcements, 1997, 3: 121-125.

[8]

Cui-Ping Cheng, Ruo-Fan An. Global stability of traveling wave fronts in a two-dimensional lattice dynamical system with global interaction. Electronic Research Archive, 2021, 29 (5) : 3535-3550. doi: 10.3934/era.2021051
[9]

Liang Zhang, Bingtuan Li. Traveling wave solutions in an integro-differential competition model. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 417-428. doi: 10.3934/dcdsb.2012.17.417
[10]

Changbing Hu, Yang Kuang, Bingtuan Li, Hao Liu. Spreading speeds and traveling wave solutions in cooperative integral-differential systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1663-1684. doi: 10.3934/dcdsb.2015.20.1663
[11]

Shiwang Ma, Xiao-Qiang Zhao. Global asymptotic stability of minimal fronts in monostable lattice equations. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 259-275. doi: 10.3934/dcds.2008.21.259
[12]

Jong-Shenq Guo, Chang-Hong Wu. Front propagation for a two-dimensional periodic monostable lattice dynamical system. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 197-223. doi: 10.3934/dcds.2010.26.197
[13]

Peter Bates, Chunlei Zhang. Traveling pulses for the Klein-Gordon equation on a lattice or continuum with long-range interaction. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 235-252. doi: 10.3934/dcds.2006.16.235
[14]

Hirokazu Ninomiya. Entire solutions and traveling wave solutions of the Allen-Cahn-Nagumo equation. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2001-2019. doi: 10.3934/dcds.2019084
[15]

Lijun Zhang, Peiying Yuan, Jingli Fu, Chaudry Masood Khalique. Bifurcations and exact traveling wave solutions of the Zakharov-Rubenchik equation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2927-2939. doi: 10.3934/dcdss.2020214
[16]

Xiaojie Hou, Yi Li, Kenneth R. Meyer. Traveling wave solutions for a reaction diffusion equation with double degenerate nonlinearities. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 265-290. doi: 10.3934/dcds.2010.26.265
[17]

Bendong Lou. Traveling wave solutions of a generalized curvature flow equation in the plane. Conference Publications, 2007, 2007 (Special) : 687-693. doi: 10.3934/proc.2007.2007.687
[18]

Faustino Sánchez-Garduño, Philip K. Maini, Judith Pérez-Velázquez. A non-linear degenerate equation for direct aggregation and traveling wave dynamics. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 455-487. doi: 10.3934/dcdsb.2010.13.455
[19]

Anna Geyer, Ronald Quirchmayr. Traveling wave solutions of a highly nonlinear shallow water equation. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1567-1604. doi: 10.3934/dcds.2018065
[20]

Aiyong Chen, Chi Zhang, Wentao Huang. Limit speed of traveling wave solutions for the perturbed generalized KdV equation. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022048

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (57)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy