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

Aaron Hoffman; Benjamin Kennedy

doi:10.3934/dcds.2011.30.137

Article Contents

2011, Volume 30, Issue 1: 137-167. Doi: 10.3934/dcds.2011.30.137

This issue Previous Article Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent Next Article Existence of multiple spike stationary patterns in a chemotaxis model with weak saturation

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
2.
Gettysburg College, Department of Mathematics, 300 North Washington St., Gettysburg, PA 17325-1400

Received: October 2009

Revised: November 2010

Published: January 2011

Abstract / Introduction Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 37L60, 34K10; Secondary: 34C19, 34K19.

Citation:

Related Papers

Cited by

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, Translations of Mathematical Monographs, Vol. 43, American Mathematical Society, Providence, R.I.
[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.