A geometric analysis of front propagation in an integrable Nagumo equation with a linear cut-off

Physica D: Nonlinear Phenomena

Volumn 241, Issue 22, 2012, Pages 1976-1984

  Popović, Nikola ^a  

^a UNIVERSITY OF EDINBURGH   (United Kingdom)

Author keywords

Cut offs; Front propagation; Geometric desingularization; Maxwell point; Reaction diffusion equations

Indexed keywords

DYNAMICAL SYSTEMS; GEOMETRY; LINEAR EQUATIONS;

CUT-OFFS; FRONT PROPAGATION; GEOMETRIC DESINGULARIZATION; MAXWELL POINTS; REACTION DIFFUSION EQUATIONS;

MAXWELL EQUATIONS;

EID: 84868193595     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2011.05.007     Document Type: Article

References (21)

1. J. Keener, and J. Sneyd Mathematical Physiology Interdisciplinary Applied Mathematics vol. 8 1998 Springer-Verlag New York
2. R. Benguria, M. Depassier, and V. Haikala Effect of a cutoff on pushed and bistable fronts of the reaction-diffusion equation Phys. Rev. E 76 5 2007 051101
3. F. Dumortier, N. Popović, and T. Kaper A geometric approach to bistable front propagation in scalar reaction-diffusion equations with cut-off Physica D 239 20 2010 1984 1999
4. N. Popović, A geometric analysis of front propagation in a family of degenerate reaction-diffusion equations with cut-off, Z. Angew. Math. Phys. (2011), in press (doi:10.1007/s00033-011-0115-6).
5. E. Brunet, and B. Derrida Shift in the velocity of a front due to a cut-off Phys. Rev. E 56 3 2007 2597 2604
6. R. Benguria, and M. Depassier Speed of fronts of the reaction-diffusion equation Phys. Rev. Lett. 77 6 1996 1171 1173
7. R. Benguria, and M. Depassier Speed of pulled fronts with a cutoff Phys. Rev. E 75 5 2007 051106
8. D. Kessler, Z. Ner, and L. Sander Front propagation: precursors, cutoffs, and structural stability Phys. Rev. E 58 1 1998 107 114
9. V. Méndez, D. Campos, and E. Zemskov Variational principles and the shift in the front speed due to a cutoff Phys. Rev. E 72 5 2005 056113
10. F. Dumortier, N. Popović, and T. Kaper The critical wave speed for the Fisher-Kolmogorov-Petrowskii-Piscounov equation with cut-off Nonlinearity 20 4 2007 855 877
11. F. Dumortier Techniques in the theory of local bifurcations: blow-up, normal forms, nilpotent bifurcations, singular perturbations D. Schlomiuk, Bifurcations and Periodic Orbits of Vector Fields NATO ASI Series C, Mathematical and Physical Sciences vol. 408 1993 Kluwer Academic Publishers Dordrecht, The Netherlands 19 73
12. F. Dumortier, and R. Roussarie Canard cycles and center manifolds Mem. Amer. Math. Soc. 121 577 1996
13. M. Krupa, and P. Szmolyan Extending geometric singular perturbation theory to nonhyperbolic points - fold and canard points in two dimensions SIAM J. Math. Anal. 33 2 2001 286 314
14. M. Hirsch, C. Pugh, and M. Shub Invariant manifolds Lecture Notes in Mathematics vol. 583 1977 Springer-Verlag Berlin
15. L. Perko Differential Equations and Dynamical Systems 3rd Edition Texts in Applied Mathematics vol. 7 2001 Springer-Verlag New York
16. B. Gilding, and R. Kersner Travelling Waves in Nonlinear Diffusion-Convection Reaction Progress in Nonlinear Differential Equations and their Applications vol. 60 2004 Birkhäuser Verlag Basel
17. N. Fenichel Geometric singular perturbation theory for ordinary differential equations J. Differential Equations 31 1 1979 53 98
18. A. Engel Noise-induced front propagation in a bistable system Phys. Lett. A 113 3 1985 139 142
19. M. Abramowitz, and I. Stegun Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series vol. 55 1972 Dover Publications New York
20. N. Popović A geometric analysis of logarithmic switchback phenomena HAMSA 2004: Proceedings of the International Workshop on Hysteresis and Multi-Scale Asymptotics, Cork, Ireland, 2004 J. Phys. Conference Series vol. 22 2005 164 173
21. N. Popović, and T. Kaper Rigorous asymptotic expansions for critical wave speeds in a family of scalar reaction-diffusion equations J. Dynam. Differential Equations 18 1 2006 103 139