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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 & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[19]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[26]

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

[27]

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

[28]

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

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

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