Carpio Rodríguez, Ana María and Bonilla , L.L.
(2003)
*Depinning transitions in discrete reaction-diffusion equations.*
SIAM Journal on applied mathematics, 63
(3).
pp. 1056-1082.
ISSN 0036-1399

PDF
Restricted to Repository staff only until 2020. 379kB |

Official URL: http://www.siam.org/journals/siap/63-3/39006.html

## Abstract

We consider spatially discrete bistable reaction-diffusion equations that admit wave front solutions. Depending on the parameters involved, such wave fronts appear to be pinned or to glide at a certain speed. We study the transition of traveling waves to steady solutions near threshold and give conditions for front pinning (propagation failure). The critical parameter values are characterized at the depinning transition, and an approximation for the front speed just beyond threshold is given.

Item Type: | Article |
---|---|

Uncontrolled Keywords: | Discrete reaction-diffusion equations, traveling wave fronts, propagation failure, wave front depinning |

Subjects: | Sciences > Mathematics > Differential equations |

ID Code: | 15045 |

References: |
[1] A. Amann,A. Wacker,L. L. Bonilla,and E. Sch¨oll, Dynamic scenarios of multistable switching in semiconductor superlattices, Phys. Rev. E, 63 (2001), paper 066207. [2] L. L. Bonilla, Stable probability densities and phase transitions for mean-field models in the thermodynamic limit, J. Statist. Phys., 46 (1987), pp. 659–678. [3] L. L. Bonilla,J. Gal´an,J. A. Cuesta,F. C. Mart´ınez,and J. M. Molera, Dynamics of electric field domains and oscillations of the photocurrent in a simple superlattice model, Phys. Rev. B, 50 (1994), pp. 8644–8657. [4] O. M. Braun and Yu. S. Kivshar, Nonlinear dynamics of the Frenkel-Kontorova model, Phys. Rep., 306 (1998), pp. 1–108. [5] P. C. Bressloff and G. Rowlands, Exact travelling wave solutions of an “integrable” discrete reaction-diffusion equation, Phys. D, 106 (1997), pp. 255–269. [6] A. E. Bugrim,A. M. Zhabotinsky,and I. R. Epstein, Calcium waves in a model with a random spatially discrete distribution of Ca2+ release sites, Biophys. J., 73 (1997), pp. 2897–2906. [7] J. W. Cahn, Theory of crystal growth and interface motion in crystalline materials, Acta Metallurgica, 8 (1960), pp. 554–562. [8] A. Carpio,S. J. Chapman,S. Hastings,and J. B. McLeod, Wave solutions for a discrete reaction-diffusion equation, European J. Appl. Math., 11 (2000), pp. 399–412. [9] A. Carpio,L. L. Bonilla,A. Wacker,and E. Sch¨oll, Wave fronts may move upstream in doped semiconductor superlattices, Phys. Rev. E, 61 (2000), pp. 4866–4876. [10] A. Carpio and L. L. Bonilla, Wave front depinning transition in discrete one-dimensional reaction-diffusion systems, Phys. Rev. Lett., 86 (2001), pp. 6034–6037. [11] A. Carpio,L. L. Bonilla,and G. Dell’Acqua, Motion of wave fronts in semiconductor superlattices, Phys. Rev. E, 64 (2001), paper 036204. [12] A. Carpio,L. L. Bonilla,and A. Luz´on, Effects of disorder on the wave front depinning transition in spatially discrete systems, Phys. Rev. E, 65 (2002), paper 035207(R) [13] P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics, Cambridge University Press, Cambridge, UK, 1995. [14] T. Erneux and G. Nicolis, Propagating waves in discrete reaction-diffusion systems, Phys. D, 67 (1993), pp. 237–244. [15] G. F´ath, Propagation failure of traveling waves in a discrete bistable medium, Phys. D, 116 (1998), pp. 176–190. [16] S. Flach,Y. Zolotaryuk,and K. Kladko, Moving lattice kinks and pulses: An inverse method, Phys. Rev. E, 59 (1999), pp. 6105–6115. [17] J. Frenkel and T. Kontorova, On the theory of plastic deformation and twinning, J. Phys. USSR, 13 (1938), pp. 1–10. [18] E. Gerde and M. Marder, Friction and fracture, Nature, 413 (2001), pp. 285–288. [19] G. Gr¨uner, The dynamics of charge-density waves, Rev. Modern Phys., 60 (1988), pp. 1129– 1181. [20] V. Hakim and K. Mallick, Exponentially small splitting of separatrices, matching in the complex plane and Borel summation, Nonlinearity, 6 (1993), pp. 57–70. [21] R. Hobart, Peierls-barrier minima, J. Appl. Phys., 36 (1965), pp. 1948–1952. [22] V. L. Indenbom, Mobility of dislocations in the Frenkel-Kontorova model, Soviet Phys.- Crystallogr., 3 (1959), pp. 193–201 (translated from Kristallografiya, 3 (1958), pp. 197– 206). [23] J. P. Keener, Propagation and its failure in coupled systems of discrete excitable cells, SIAM J. Appl. Math., 47 (1987), pp. 556–572. [24] J. P. Keener and J. Sneyd, Mathematical Physiology, Springer, NewY ork, 1998. [25] P. G. Kevrekidis,C. K. R. T. Jones,and T. Kapitula, Exponentially small splitting of heteroclinic orbits: From the rapidly forced pendulum to discrete solitons, Phys. Lett. A, 269 (2000), pp. 120–129. [26] P. G. Kevrekidis,I. G. Kevrekidis,and A. R. Bishop, Propagation failure, universal scalings and Goldstone modes, Phys. Lett. A, 279 (2001), pp. 361–369. [27] J. R. King and S. J. Chapman, Asymptotics beyond all orders and Stokes lines in nonlinear differential-difference equations, European J. Appl. Math., 12 (2001), pp. 433–463. [28] K. Kladko,I. Mitkov,and A. R. Bishop, Universal scaling of wave propagation failure in arrays of coupled nonlinear cells, Phys. Rev. Lett., 84 (2000), pp. 4505–4508. [29] J. Miles, On Faraday resonance of a viscous liquid, J. Fluid Mech., 395 (1999), pp. 321–325. [30] I. Mitkov,K. Kladko,and J. E. Pearson, Tunable pinning of bursting waves in extended systems with discrete sources, Phys. Rev. Lett., 81 (1998), pp. 5453–5456. [31] F. R. N. Nabarro, Dislocations in a simple cubic lattice, Proc. Phys. Soc. London, 59 (1947), pp. 256–272. [32] F. R. N. Nabarro, Theory of Crystal Dislocations, Oxford University Press, Oxford, UK, 1967. [33] R. Peierls, The size of a dislocation, Proc. Phys. Soc. London, 52 (1940), pp. 34–37. [34] V. H. Schmidt, Exact solution in the discrete case for solitons propagating in a chain of harmonically coupled particles lying in double-minimum potential wells, Phys. Rev. B, 20 (1979), pp. 4397–4405. [35] L. I. Slepyan, Dynamics of a crack in a lattice, Sov. Phys. Dokl., 26 (1981), pp. 538–540 (translated from Dokl. Akad. Nauk SSSR, 258 (1981), pp. 561–564). [36] J. M. Speight and R. S. Ward, Kink dynamics in a novel discrete sine-Gordon system, Nonlinearity, 7 (1994), pp. 475–484. [37] J. M. Speight, A discrete φ4 system without a Peierls-Nabarro barrier, Nonlinearity, 10 (1997), pp. 1615–1625. [38] J. M. Speight, Topological discrete kinks, Nonlinearity, 12 (1999), pp. 1373–1387. [39] H. S. J. van der Zant,T. P. Orlando,S. Watanabe,and S. H. Strogatz, Kink propagation in a discrete system: Observation of phase locking to linear waves, Phys. Rev. Lett., 74 (1995), pp. 174–177. [40] B. Zinner, Existence of traveling wave front solutions for the discrete Nagumo equation, J. Differential Equations, 96 (1992), pp. 1–27. |

Deposited On: | 27 Apr 2012 08:46 |

Last Modified: | 06 Feb 2014 10:15 |

Repository Staff Only: item control page