Carpio Rodríguez, Ana María and Chapman, S.J. and Hastings, S. and Mcleod, J.B.
(2000)
*Wave solutions for a discrete reaction-diffusion equation.*
European Journal of Applied Mathematics, 11
(4).
pp. 399-412.
ISSN 0956-7925

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Official URL: http://journals.cambridge.org/abstract_S0956792599004222

## Abstract

Motivated by models from fracture mechanics and from biology, we study the infinite system of differential equations u'(n) = u(n-1) - 2u(n) + u(n+1) - A sin u(n) + F, ' = d/dt, where A and F are positive parameters. For fixed A > 0 we show that there are monotone travelling waves for F in an interval F-crit < F < A, and we are able to give a rigorous upper bound for F-crit, in contrast to previous work on similar problems. We raise the problem of characterizing those nonlinearities (apparently the more common) for which F-crit > 0. We show that, for the sine nonlinearity, this is true if A > 2. (Our method yields better estimates than this, but does not include all A > 0.) We also consider the existence and multiplicity of time independent solutions when \F\ < F-crit.

Item Type: | Article |
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Uncontrolled Keywords: | Nagumo equation; Systems; Propagation |

Subjects: | Sciences > Mathematics > Differential equations |

ID Code: | 15100 |

References: | Bell, J. & Cosner, C. (1984) Threshold behavior and propagation for nonlinear differential-difference systems motivated by modeling myelinated axons. Quart. J. Appl. Math. 42, 1{14. Erneux, T. & Nicolis, G. (1993) Propagating waves in discrete bistable reaction-diusion systems. Physica D, 67, 237{244. Frenkel, J. & Kontorova, T. (1938) On the theory of plastic deformation and twinning. J.Phys. USSR, 13, 1{10. Hastings, S. P. & McLeod, J. B. In preparation. Hankerson, D. & Zinner, B. (1992) Wavefronts for a cooperative tridiagonal system of differential equations. J. Dynamics & Di. Eq. 5, 359{373. Keener, J. P. (1987) Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. 47, 556{572. King, J. R. & Chapman, S. J. Asymptotics beyond all orders and Stokes lines in nonlinear differential-difference equations. Preprint. Mallet-Paret, J. (1997) The global structure of traveling waves in spatially discrete dynamical systems. Preprint, August. Walter, W. (1970) Differential and Integral Inequalities. Springer-Verlag. Zinner, B. (1992) Existence of traveling wavefront solutions for the discrete Nagumo equation. J. Di. Eq. 96, 1{27. Zinner, B. (1991) Stability of traveling wavefronts for the discrete Nagumo equation. SIAM J. Math. Anal. 22, 1016{1020. |

Deposited On: | 04 May 2012 11:49 |

Last Modified: | 06 Feb 2014 10:16 |

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