Abstract
The subject of the present chapter is the front propagation in reaction-diffusion systems. We start with the consideration of the basic model, Fisher-Kolmogorov equation, and discuss the linear and nonlinear criteria of the velocity selection for fronts between a stable and an unstable stationary states. Then we consider the dynamics of plane and curved fronts between stable stationary states. The last section of this chapter contains the description of combustion fronts and their stability.
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Bibliography
I.S. Aranson and L. Kramer. The world of the complex Ginzburg-Landau equation. Rev. Mod. Phys., 74:99, 2002.
D.G. Aronson and H.F. Weinberger. Multidimensional nonlinear diffusion arising in population genetics. Adv. Math., 30:33, 1978.
T. Bohr, M.H. Jensen, G. Paladin, and A. Vulpiani. Dynamical Systems Approach to Turbulence. Cambridge University Press, 1998.
R.A. Fisher. The wave of advance of advantageous genes. Ann. Eugenics, 7:355, 1937.
E. Knobloch and J. De Luca. Amplitude equations for travelling wave convection. Nonlinearity, 3:975, 1990.
A. Kolmogoroff, I. Petrovsky, and N. Piscounoff. Study of the diffusion equation with growth of the quantity of matter and its application to a biological problem. Moscow University, Bull. Math., 1:1, 1937.
S.B. Margolis and B.J. Matkowsky. Flame propagation in channels: Secondary bifurcation to quasi-periodic pulsations. SIAM J. Appl. Math, 45:93, 1985.
B.J. Matkowsky and G.I. Sivashinsky. Propagation of a pulsating reaction front in solid fuel combustion. SIAM J. Appl. Math, 37:686, 1979.
B.J. Matkowsky and V.A. Volpert. Coupled nonlocal complex Ginzburg-Landau equations in gasless combustion. Physica D, 54:203, 1992.
J.D. Murray. Mathematical Biology. Springer, New York, 2002.
G.I. Sivashinsky. Nonlinear analysis of hydrodynamic instability in laminar flames — I. Derivation of basic equations. Acta Astronautica, 4:1177, 1977.
P.-F. Verhulst. Notice sur la loi que la population poursuit dans son accroissement. Corr. Math. et Phys., 10:113, 1838.
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Nepomnyashchy, A.A. (2010). Nonlinear dynamics of fronts. In: Colinet, P., Nepomnyashchy, A. (eds) Pattern Formation at Interfaces. CISM International Centre for Mechanical Sciences, vol 513. Springer, Vienna. https://doi.org/10.1007/978-3-7091-0125-4_2
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DOI: https://doi.org/10.1007/978-3-7091-0125-4_2
Publisher Name: Springer, Vienna
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