Fronts in lattices

Thumbnail Image
Official URL
Full text at PDC
Publication Date
Advisors (or tutors)
Journal Title
Journal ISSN
Volume Title
Google Scholar
Research Projects
Organizational Units
Journal Issue
Simple models of defect motion in lattices identify dislocations [1] and cracks [8,9] with discrete traveling waves [4,10]. In overdamped limits, such lattice models often become discrete bistable equations [3,5], similar to the ones encountered in biology (to describe nerve propagation [7], for instance). In this talk, we will review recent results on front propagation in spatially discrete models, discussing existence of stationary and travelling wave fronts [2,5,6] together with strategies to predict their speeds and the thresholds for propagation failure [3].
[1] A Carpio, SJ Chapman, SD Howison, JR Ockendon, Dynamics of line singularities, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 355(1731), 2013-2024, 1997 [2] A Carpio, SJ Chapman, S Hastings, JB McLeod, Wave solutions for a discrete reaction-diffusion equation, European Journal of Applied Mathematics 11 (4), 399-412, 2000 [3] A Carpio, LL Bonilla, Depinning transitions in discrete reaction-diffusion equations, SIAM Journal on Applied Mathematics 63 (3), 1056-1082, 2003 [4] A Carpio, LL Bonilla, Edge dislocations in crystal structures considered as traveling waves in discrete models, Physical Review Letters 90 (13), 135502, 2003 [5] A Carpio, LL Bonilla, Oscillatory wave fronts in chains of coupled nonlinear oscillators, Physical Review E 67 (5), 056621, 2003 [6] A Carpio, Nonlinear stability of oscillatory wave fronts in chains of coupled oscillators, Physical Review E 69 (4), 046601, 2004 [7] A Carpio, Asymptotic construction of pulses in the discrete Hodgkin-Huxley model for myelinated nerves, Physical Review E 72 (1), 011905, 2005 [8] I Plans, A Carpio, LL Bonilla, Homogeneous nucleation of dislocations as bifurcations in a periodized discrete elasticity model, EPL (Europhysics Letters) 81 (3), 36001, 2008 [9] I Plans, A Carpio, LL Bonilla, Toy nanoindentation model and incipient plasticity, Chaos, Solitons & Fractals 42 (3), 1623-1630, 2009 [10] LL Bonilla, A Carpio, A Prados, RR Rosales, Ripples in a string coupled to Glauber spins, Physical Review E 85 (3), 031125, 2012