Competing interactions and traveling wave solutions in lattice differential equations. (English) Zbl 1331.39005
Summary: 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 [P. W. Bates et al., SIAM J. Math. Anal. 35, No. 2, 520–546 (2003; Zbl 1050.37041)], 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.
MSC:
| 39A12 | Discrete version of topics in analysis |
| 34K31 | Lattice functional-differential equations |
| 35K57 | Reaction-diffusion equations |
| 37L60 | Lattice dynamics and infinite-dimensional dissipative dynamical systems |
Keywords:
bistable; traveling waves; competing interaction; Fredholm operator; lattice differential equationCitations:
Zbl 1050.37041References:
| [1] | P. W. Bates, Traveling waves of bistable dynamics on a lattice,, SIAM J. Math. Anal., 35, 520 (2003) · Zbl 1050.37041 · doi:10.1137/S0036141000374002 |
| [2] | M. Brucal - Hallare, Traveling fronts in an antidiffusion lattice Nagumo model,, SIAM J. Appl. Dyn. Sys., 10, 921 (2011) · Zbl 1236.39005 · doi:10.1137/100819461 |
| [3] | J. W. Cahn, Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice,, SIAM J. Appl. Math., 59, 455 (1999) · Zbl 0917.34052 · doi:10.1137/S0036139996312703 |
| [4] | X. F. Chen, Traveling waves in discrete periodic media for bistable dynamics,, Arch. Ration. Mech. Anal., 189, 189 (2008) · Zbl 1152.37033 · doi:10.1007/s00205-007-0103-3 |
| [5] | S. N. Chow, Traveling waves in lattice dynamical systems,, J. Differential Equations, 149, 248 (1998) · Zbl 0911.34050 · doi:10.1006/jdeq.1998.3478 |
| [6] | J. Harterich, Exponential dichotomies for linear non-autonomous functional differential equations of mixed type,, Indiana Univ. Math. J., 51, 1081 (2002) · Zbl 1047.34079 · doi:10.1512/iumj.2002.51.2188 |
| [7] | H. J. Hupkes, Analysis of Newton’s method to compute travelling waves in discrete media,, J. Dynam. Differential Equations, 17, 523 (2005) · Zbl 1144.34382 · doi:10.1007/s10884-005-5809-z |
| [8] | H. J. Hupkes, Center manifold theory for functional differential equations of mixed type,, J. Dynam. Diff. Eqns., 19, 497 (2007) · Zbl 1132.34054 · doi:10.1007/s10884-006-9055-9 |
| [9] | H. J. Hupkes, Lin’s method and homoclinic bifurcations for functional differential equations of mixed type,, Indiana Univ. Math. J., 58, 2433 (2009) · Zbl 1214.34058 · doi:10.1512/iumj.2009.58.3661 |
| [10] | H. J. Hupkes, Traveling pulse solutions for the discrete FitzHugh-Nagumo system,, SIAM J. Appl. Dyn. Syst., 9, 827 (2010) · Zbl 1211.34007 · doi:10.1137/090771740 |
| [11] | H. J. Hupkes, Negative diffusion and traveling waves in high dimensional lattice systems,, SIAM J. Math. Anal., 45, 1068 (2013) · Zbl 1301.34098 · doi:10.1137/120880628 |
| [12] | C. Lamb, Neutral mixed type functional differential equations,, J. Dynam. Differential Equations (2015) |
| [13] | J. Mallet-Paret, Exponential dichotomies and Wiener-Hopf factorizations for mixed-type functional differential equations,, J. of Differential Equations · Zbl 0927.34049 |
| [14] | J. Mallet-Paret, The Fredholm alternative for functional differential equations of mixed type,, J. Dynamics and Differential Equations, 11, 1 (1999) · Zbl 0927.34049 · 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, 49 (1999) · Zbl 0921.34046 · doi:10.1023/A:1021841618074 |
| [16] | J. Mallet-Paret, Traveling waves in spatially-discrete dynamical systems of diffusive type,, Lecture Notes in Math, 1822, 231 (2003) · Zbl 1083.37058 · 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, 121 (1989) · Zbl 0684.34065 · doi:10.1007/BF01047828 |
| [18] | A. Rustichini, Hopf bifurcation for functional-differential equations of mixed type,, J. Dynam. Differential Equations, 1, 145 (1989) · Zbl 0684.34070 · 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, 1 (1999) · Zbl 0939.35016 · 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, 55 (1999) · Zbl 0939.35015 · doi:10.1006/jdeq.1999.3652 |
| [21] | A. Vainchtein, Nucleation and propagation of phase mixtures in a bistable chain,, Phys. Rev. B., 79 (2009) |
| [22] | A. Vainchtein, Propagation of periodic patterns in a discrete system with competing interactions,, SIAM J. Appl. Dyn. Sys., 14, 523 (2015) · Zbl 1387.34108 |
| [23] | B. Zinner, Stability of traveling wavefronts for the discrete Nagumo equation,, SIAM J. Math. Anal., 22, 1016 (1991) · Zbl 0739.34060 · doi:10.1137/0522066 |
| [24] | B. Zinner, Existence of traveling wavefront solutions for the discrete Nagumo equation,, J. Diff. Eqn, 96, 1 (1992) · Zbl 0752.34007 · doi:10.1016/0022-0396(92)90142-A |
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