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Carpio, Ana 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

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http://www.cambridge.org/ | Publisher |

## 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 |

Deposited On: | 04 May 2012 11:49 |

Last Modified: | 12 Dec 2018 15:07 |

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