Research on Nonlinear waves

Fronts and pulses propagating in lattices seem to describe physical reality in many different fields: atoms adsorbed on a periodic substrate, motion of dislocations in crystals, propagation of cracks in brittle materials, domain dynamics in semiconductor superlattices, nerve impulse propagation along myelinated nerves, calcium release waves… We have analyzed travelling waves propagating in nonlinear lattices, characterizing pinning and propagation failure in different physical and biological systems by means of bifurcations and active point approximations, which yield predictions of wave profiles and wave speeds by asymptotic methods, even in the presence of stochastic noise. Aplications in biological sciences and materials science are discussed in the corresponding sections.
Additional work in this field include the introduction of nonreflecting boundary conditions for discrete waves, blow up studies and the analysis of aggregation models for bubbles in radioactive waste.

Positivity preserving high order schemes for kinetic models of angiogenesis, A. Carpio, E. Cebrián, International Journal on Nonlinear Sciences and Numerical Simulation 23(6), 917-929, 2022 [pdf] [arxiv]

A convergent numerical scheme for integrodifferential kinetic models of angiogenesis, L.L. Bonilla, A. Carpio, M. Carretero, G. Duro, M. Negreanu, F. Terragni, Journal of Computational Physics 375, 1270-1294, 2018 [pdf]

Propagation failure along myelinated nerves, A. Carpio, I. Peral, Journal of Nonlinear Science 21(4), 499-520, 2011 [pdf] [archivo]

Analysis of helium bubble growth in radioactive waste, A. Carpio, B. Tapiador, Nonlinear Analysis-Real World Applications 11(5), 4174-4184, 2010 [pdf] [archivo]

Nonreflecting boundary conditions for discrete waves, A. Carpio, B. Tapiador, Journal of Computational Physics 229(5), 1879-1896, 2010 [pdf] [archivo]

Explosive behaviour in spatially discrete reaction-diffusion systems, A. Carpio, G. Duro, Discrete and Continuous Dynamical Systems - Series B 12(4), 693-711, 2009 [pdf] [archivo]

Self-sustained current oscillations in the kinetic theory of semiconductor superlattices, E. Cebrián, L.L. Bonilla, A. Carpio, Journal of Computational Physics 228(20), 7689-7705, 2009 [pdf] [arxiv]

Kinetics of helium bubble formation in nuclear materials, L.L. Bonilla, A. Carpio, J.C. Neu, W.G. Wolfer, Physica D- Nonlinear Phenomena 222(1-2), 131-140, 2006 [pdf] [arxiv]

Asymptotic construction of pulses in the discrete Hodgkin-Huxley model for myelinated nerves, A. Carpio, Physical Review E 72(1), 011905, 2005 [pdf] [arxiv]

Instability and collapse in discrete wave equations, A. Carpio, G. Duro, Computational Methods in Applied Mathematics, 5(3), 223-241, 2005 [pdf]

Wave trains, self-oscillations and synchronization in discrete media, A. Carpio, Physica D-Nonlinear Phenomena 207(1-2), 117-136, 2005 [pdf] [arxiv]

Nonlinear stability of oscillatory wave fronts in chains of coupled oscillators, A. Carpio, Physical Review E 69(4), 046601, 2004 [pdf] [arxiv]

Pulse propagation in discrete systems of coupled excitable cells, A. Carpio, LL Bonilla, SIAM Journal on Applied Mathematics 63(2), 619-635, 2003 [pdf] [arxiv]

Edge dislocations in crystal structures considered as travelling waves in discrete models, A. Carpio, L.L. Bonilla, Physical Review Letters 90(13), 135502, 2003; 91(2), 029901 2003 [pdf1] [pdf2] [arxiv]

Oscillatory wave fronts in chains of coupled nonlinear oscillators, A. Carpio, L.L. Bonilla, Physical Review E 67(5), 056621, 2003 [pdf] [arxiv]

Depinning transitions in discrete reaction-diffusion equations, A. Carpio, L.L. Bonilla, SIAM Journal on Applied Mathematics 63(3), 1056-1082, 2003 [pdf] [arxiv]

Effects of disorder on the wave front depinning transition in spatially discrete systems, A. Carpio, L.L. Bonilla, A. Luzon, Physical Review E (RC) 65(3), 035207, 2002 [pdf] [arxiv]

Wavefronts for discrete two-dimensional nonlinear diffusion equations, A. Carpio, Applied Mathematics Letters 15(4), 415-421, 2002 [pdf] [archivo]

Wave front depinning transition in discrete one-dimensional reaction-diffusion systems, A. Carpio, LL Bonilla, Physical Review Letters 86(26), 6034-6037, 2001 [pdf] [arxiv]

Numerical study of hyperbolic equations with integral constraints arising in semiconductor theory, A. Carpio, P. Hernando, M. Kindelan, SIAM Journal on Numerical Analysis 39(1), 168-191, 2001 [pdf] [arxiv]

Motion of wave fronts in semiconductor superlattices, A. Carpio, LL Bonilla, G. Dell'Acqua, Physical Review E 64(3), 036204, 2001 [pdf] [arxiv]

Wave solutions for a discrete reaction-diffusion equation, A. Carpio, SJ Chapman, S Hastings, JB Mcleod, European Journal of Applied Mathematics 11(4), 399-412, 2000 [pdf] [archivo]

Wave fronts may move upstream in semiconductor superlattices, A. Carpio, LL Bonilla, A Wacker, E Schöll, Physical Review E 61(5), 4866-4876, 2000 [pdf] [arxiv]