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Wave solutions for a discrete reaction-diffusion equation

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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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
Uncontrolled Keywords:Nagumo equation; Systems; Propagation
Subjects:Sciences > Mathematics > Differential equations
ID Code:15100
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Deposited On:04 May 2012 11:49
Last Modified:06 Feb 2014 10:16

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