American Institute of Mathematical Sciences

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March  2016, 15(2): 457-475. doi: 10.3934/cpaa.2016.15.457

Competing interactions and traveling wave solutions in lattice differential equations

E. S. Van Vleck 1, and  Aijun Zhang 2,

1. 

Department of Mathematics, University of Kansas, Lawrence, KS 66045, United States
2. 

Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR 72701, United States

Received  March 2015 Revised  November 2015 Published  January 2016

The existence of traveling front solutions to bistable lattice differential equations in the absence of a comparison principle is studied. The results are in the spirit of those in Bates, Chen, and Chmaj [1], but are applicable to vector equations and to more general limiting systems. An abstract result on the persistence of traveling wave solutions is obtained and is then applied to lattice differential equations with repelling first and/or second neighbor interactions and to some problems with infinite range interactions.
Keywords: Bistable, traveling waves, Fredholm operator, competing interaction, lattice differential equation..
Mathematics Subject Classification: Primary: 39A12, 34K31, 35K57, 37L6.
Citation: 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
References:
[1]

P. W. Bates, X. F. 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]

M. Brucal - Hallare and E.S. Van Vleck, Traveling fronts in an antidiffusion lattice Nagumo model, SIAM J. Appl. Dyn. Sys., 10 (2011), 921-959. doi: 10.1137/100819461.

[3]

J. W. Cahn, J. Mallet-Paret and E. S. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Appl. Math., 59 (1999), 455-493. doi: 10.1137/S0036139996312703.

[4]

X. F. Chen, J. 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.

[5]

S. N. Chow, J. Mallet-Paret and W. X. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291. doi: 10.1006/jdeq.1998.3478.

[6]

J. Harterich, B. Sandstede and A. Scheel, Exponential dichotomies for linear non-autonomous functional differential equations of mixed type, Indiana Univ. Math. J., 51 (2002), 1081-1109. doi: 10.1512/iumj.2002.51.2188.

[7]

H. J. Hupkes and S. M. Verduyn-Lunel, Analysis of Newton's method to compute travelling waves in discrete media, J. Dynam. Differential Equations, 17 (2005), 523-572. doi: 10.1007/s10884-005-5809-z.

[8]

H. J. Hupkes and S. M. Verduyn-Lunel, Center manifold theory for functional differential equations of mixed type, J. Dynam. Diff. Eqns., 19 (2007), 497-560. doi: 10.1007/s10884-006-9055-9.

[9]

H. J. Hupkes and S. M. Verduyn-Lunel, Lin's method and homoclinic bifurcations for functional differential equations of mixed type, Indiana Univ. Math. J., 58 (2009), 2433-2487. doi: 10.1512/iumj.2009.58.3661.

[10]

H. J. Hupkes and B. Sandstede, Traveling pulse solutions for the discrete FitzHugh-Nagumo system, SIAM J. Appl. Dyn. Syst., 9 (2010), 827-882. doi: 10.1137/090771740.

[11]

H. J. Hupkes and E. S. Van Vleck, Negative diffusion and traveling waves in high dimensional lattice systems, SIAM J. Math. Anal., 45 (2013), 1068-1135. doi: 10.1137/120880628.

[12]

C. Lamb and E. S. Van Vleck, Neutral mixed type functional differential equations, J. Dynam. Differential Equations, (2015), in press.

[13]

J. Mallet-Paret and S. M. Verduyn-Lunel, Exponential dichotomies and Wiener-Hopf factorizations for mixed-type functional differential equations,, \emph{J. of Differential Equations}, (). 

[14]

J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type, J. Dynamics and Differential Equations, 11 (1999), 1-48. doi: 10.1023/A:1021889401235.

[15]

J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems, J. Dyn. Diff. Eqn., 11 (1999), 49-127. doi: 10.1023/A:1021841618074.

[16]

J. Mallet-Paret, Traveling waves in spatially-discrete dynamical systems of diffusive type, Lecture Notes in Math, 1822 (2003), 231-298. doi: 10.1007/978-3-540-45204-1_4.

[17]

A. Rustichini, Functional-differential equations of mixed type: the linear autonomous case, J. Dynam. Differential Equations, 1 (1989), 121-143, doi: 10.1007/BF01047828.

[18]

A. Rustichini, Hopf bifurcation for functional-differential equations of mixed type, J. Dynam. Differential Equations, 1 (1989), 145-177. doi: 10.1007/BF01047829.

[19]

W. Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities: I. Stability and uniqueness, J. Differential Equations, 159 (1999), 1-54. doi: 10.1006/jdeq.1999.3651.

[20]

W. Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities: II. Existence, J. Differential Equations, 159 (1999), 55-101. doi: 10.1006/jdeq.1999.3652.

[21]

A. Vainchtein and E. S. Van Vleck, Nucleation and propagation of phase mixtures in a bistable chain, Phys. Rev. B., 79 (2009), 144123.

[22]

A. Vainchtein, E. S. Van Vleck and A. Zhang, Propagation of periodic patterns in a discrete system with competing interactions, SIAM J. Appl. Dyn. Sys., 14 (2015), 523-555.

[23]

B. Zinner, Stability of traveling wavefronts for the discrete Nagumo equation, SIAM J. Math. Anal., 22 (1991), 1016-1020. doi: 10.1137/0522066.

[24]

B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Diff. Eqn, 96 (1992), 1-27. doi: 10.1016/0022-0396(92)90142-A.

show all references

References:
[1]

P. W. Bates, X. F. 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]

M. Brucal - Hallare and E.S. Van Vleck, Traveling fronts in an antidiffusion lattice Nagumo model, SIAM J. Appl. Dyn. Sys., 10 (2011), 921-959. doi: 10.1137/100819461.

[3]

J. W. Cahn, J. Mallet-Paret and E. S. Van Vleck, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice, SIAM J. Appl. Math., 59 (1999), 455-493. doi: 10.1137/S0036139996312703.

[4]

X. F. Chen, J. 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.

[5]

S. N. Chow, J. Mallet-Paret and W. X. Shen, Traveling waves in lattice dynamical systems, J. Differential Equations, 149 (1998), 248-291. doi: 10.1006/jdeq.1998.3478.

[6]

J. Harterich, B. Sandstede and A. Scheel, Exponential dichotomies for linear non-autonomous functional differential equations of mixed type, Indiana Univ. Math. J., 51 (2002), 1081-1109. doi: 10.1512/iumj.2002.51.2188.

[7]

H. J. Hupkes and S. M. Verduyn-Lunel, Analysis of Newton's method to compute travelling waves in discrete media, J. Dynam. Differential Equations, 17 (2005), 523-572. doi: 10.1007/s10884-005-5809-z.

[8]

H. J. Hupkes and S. M. Verduyn-Lunel, Center manifold theory for functional differential equations of mixed type, J. Dynam. Diff. Eqns., 19 (2007), 497-560. doi: 10.1007/s10884-006-9055-9.

[9]

H. J. Hupkes and S. M. Verduyn-Lunel, Lin's method and homoclinic bifurcations for functional differential equations of mixed type, Indiana Univ. Math. J., 58 (2009), 2433-2487. doi: 10.1512/iumj.2009.58.3661.

[10]

H. J. Hupkes and B. Sandstede, Traveling pulse solutions for the discrete FitzHugh-Nagumo system, SIAM J. Appl. Dyn. Syst., 9 (2010), 827-882. doi: 10.1137/090771740.

[11]

H. J. Hupkes and E. S. Van Vleck, Negative diffusion and traveling waves in high dimensional lattice systems, SIAM J. Math. Anal., 45 (2013), 1068-1135. doi: 10.1137/120880628.

[12]

C. Lamb and E. S. Van Vleck, Neutral mixed type functional differential equations, J. Dynam. Differential Equations, (2015), in press.

[13]

J. Mallet-Paret and S. M. Verduyn-Lunel, Exponential dichotomies and Wiener-Hopf factorizations for mixed-type functional differential equations,, \emph{J. of Differential Equations}, (). 

[14]

J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type, J. Dynamics and Differential Equations, 11 (1999), 1-48. doi: 10.1023/A:1021889401235.

[15]

J. Mallet-Paret, The global structure of traveling waves in spatially discrete dynamical systems, J. Dyn. Diff. Eqn., 11 (1999), 49-127. doi: 10.1023/A:1021841618074.

[16]

J. Mallet-Paret, Traveling waves in spatially-discrete dynamical systems of diffusive type, Lecture Notes in Math, 1822 (2003), 231-298. doi: 10.1007/978-3-540-45204-1_4.

[17]

A. Rustichini, Functional-differential equations of mixed type: the linear autonomous case, J. Dynam. Differential Equations, 1 (1989), 121-143, doi: 10.1007/BF01047828.

[18]

A. Rustichini, Hopf bifurcation for functional-differential equations of mixed type, J. Dynam. Differential Equations, 1 (1989), 145-177. doi: 10.1007/BF01047829.

[19]

W. Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities: I. Stability and uniqueness, J. Differential Equations, 159 (1999), 1-54. doi: 10.1006/jdeq.1999.3651.

[20]

W. Shen, Travelling waves in time almost periodic structures governed by bistable nonlinearities: II. Existence, J. Differential Equations, 159 (1999), 55-101. doi: 10.1006/jdeq.1999.3652.

[21]

A. Vainchtein and E. S. Van Vleck, Nucleation and propagation of phase mixtures in a bistable chain, Phys. Rev. B., 79 (2009), 144123.

[22]

A. Vainchtein, E. S. Van Vleck and A. Zhang, Propagation of periodic patterns in a discrete system with competing interactions, SIAM J. Appl. Dyn. Sys., 14 (2015), 523-555.

[23]

B. Zinner, Stability of traveling wavefronts for the discrete Nagumo equation, SIAM J. Math. Anal., 22 (1991), 1016-1020. doi: 10.1137/0522066.

[24]

B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation, J. Diff. Eqn, 96 (1992), 1-27. doi: 10.1016/0022-0396(92)90142-A.

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