# American Institute of Mathematical Sciences

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

## Competing interactions and traveling wave solutions in lattice differential equations

 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.
Citation: E. S. Van Vleck, Aijun Zhang. Competing interactions and traveling wave solutions in lattice differential equations. Communications on Pure & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] C. Lamb and E. S. Van Vleck, Neutral mixed type functional differential equations, J. Dynam. Differential Equations, (2015), in press. Google Scholar [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}, ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] A. Rustichini, Hopf bifurcation for functional-differential equations of mixed type, J. Dynam. Differential Equations, 1 (1989), 145-177. doi: 10.1007/BF01047829.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] A. Vainchtein and E. S. Van Vleck, Nucleation and propagation of phase mixtures in a bistable chain, Phys. Rev. B., 79 (2009), 144123. Google Scholar [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. Google Scholar [23] B. Zinner, Stability of traveling wavefronts for the discrete Nagumo equation, SIAM J. Math. Anal., 22 (1991), 1016-1020. doi: 10.1137/0522066.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] C. Lamb and E. S. Van Vleck, Neutral mixed type functional differential equations, J. Dynam. Differential Equations, (2015), in press. Google Scholar [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}, ().   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [18] A. Rustichini, Hopf bifurcation for functional-differential equations of mixed type, J. Dynam. Differential Equations, 1 (1989), 145-177. doi: 10.1007/BF01047829.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] A. Vainchtein and E. S. Van Vleck, Nucleation and propagation of phase mixtures in a bistable chain, Phys. Rev. B., 79 (2009), 144123. Google Scholar [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. Google Scholar [23] B. Zinner, Stability of traveling wavefronts for the discrete Nagumo equation, SIAM J. Math. Anal., 22 (1991), 1016-1020. doi: 10.1137/0522066.  Google Scholar [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.  Google Scholar
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