Impacto
Carpio Rodríguez, Ana María and Bonilla, L.L. (2003) Depinning transitions in discrete reactiondiffusion equations. SIAM Journal on applied mathematics, 63 (3). pp. 10561082. ISSN 00361399

PDF
379kB 
Official URL: http://www.siam.org/journals/siap/633/39006.html
Abstract
We consider spatially discrete bistable reactiondiffusion 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 reactiondiffusion 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 meanfield 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 FrenkelKontorova model, Phys. Rep., 306 (1998), pp. 1–108. [5] P. C. Bressloff and G. Rowlands, Exact travelling wave solutions of an “integrable” discrete reactiondiffusion 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 reactiondiffusion 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 onedimensional reactiondiffusion 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 reactiondiffusion 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 chargedensity 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, Peierlsbarrier minima, J. Appl. Phys., 36 (1965), pp. 1948–1952. [22] V. L. Indenbom, Mobility of dislocations in the FrenkelKontorova 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 differentialdifference 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 doubleminimum 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 sineGordon system, Nonlinearity, 7 (1994), pp. 475–484. [37] J. M. Speight, A discrete φ4 system without a PeierlsNabarro 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:  20 Apr 2015 11:57 
Repository Staff Only: item control page