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Stability of interacting traveling waves in reactionconvectiondiffusion systems
1.  Universidade Federal de Juiz de Fora, Juiz de Fora, MG 36036900, Brazil 
2.  Columbia College, Columbia University, New York, NY 10027, United States 
3.  Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, RJ 22460320, Brazil 
4.  Department of Mathematics, North Carolina State University, Raleigh, NC 276958205 
The paper is illustrated through a simple model for the injection of air into a porous medium that contains a solid fuel. The model considered here reproduces a variety of observed phenomena and yet is simple enough to allow rigorous investigation. We refer on earlier work containing proofs of existence of traveling waves corresponding to combustion waves by phase plane analysis were presented; wave sequences that can occur as solutions of Riemann problems were identified.
References:
[1] 
I. Akkutlu and Y. Yortsos, The dynamics of insitu combustion fronts in porous media,, J. of Combustion and Flame, 134 (2003), 229. 
[2] 
A. Aldushin and S. Kasparyan, Stability of stationary filtrational combustion waves,, Combustion, 17 (1981), 615. 
[3] 
A. Aldushin, I. Rumanov and B. Matkowsky, Maximal energy accumulation in a superadiabatic filtration combustion wave,, J. of Combustion and Flame, 118 (1999), 76. 
[4] 
A. Bayliss, G. Leaf and B. Matkowsky, Pulsating and chaotic dynamics near the extinction limit,, Combustion Science and Technology, 84 (1992), 253. 
[5] 
A. Bayliss and B. Matkowsky, Two routes to chaos in condensed phase combustion,, SIAM J. Appl. Math., 50 (1990), 437. 
[6] 
A. Bayliss and B. Matkowsky, From traveling waves to chaos in combustion,, SIAM Journal on Applied Mathematics, 54 (1994), 147. 
[7] 
I. Brailovsky and G. Sivashinsky, Chaotic dynamics in solid fuel combustion,, Physica D: Nonlinear Phenomena, 65 (1993), 191. 
[8] 
J. Bruining, A. Mailybaev and D. Marchesin, Filtration combustion in wet porous medium,, SIAM Journal on Applied Mathematics, 70 (2009), 1157. 
[9] 
G. Chapiro, L. Furtado, D. Marchesin and S. Schecter, Numerical analysis of combustion waves and riemann solutions in light porous foam, 2014,, Preprint at http://preprint.impa.br/visualizar?id=5912., (). 
[10] 
G. Chapiro, A. A. Mailybaev, A. Souza, D. Marchesin and J. Bruining, Asymptotic approximation of longtime solution for lowtemperature filtration combustion,, Comput. Geosciences, 16 (2012), 799. 
[11] 
G. Chapiro, D. Marchesin and S. Schecter, Combustion waves and Riemann solutions in light porous foam,, Journal of Hyperbolic Differential Equations, 11 (2014), 295. 
[12] 
M. Decker and D. Schult, Dynamics of smoulder waves near extinction,, Combustion Theory and Modelling, 8 (2004), 491. 
[13] 
A. Ghazaryan, Y. Latushkin, S. Schecter and A. de Souza, Stability of gasless combustion fronts in onedimensional solids,, Archive for Rational Mechanics and Analysis, 198 (2010), 981. 
[14] 
L. W. Lake, Enhanced oil recovery,, Old Tappan, (1989). 
[15] 
A. Mailybaev, J. Bruining and D. Marchesin, Analysis of in situ combustion of oil with pyrolysis and vaporization,, Combustion and Flame, 158 (2011), 1097. 
[16] 
A. Mailybaev, D. Marchesin and J. Bruining, Resonance in lowtemperature oxidation waves for porous media,, SIAM Journal on Mathematical Analysis, 43 (2011), 2230. 
[17] 
D. Marchesin and S. Schecter, Oxidation heat pulses in twophase expansive flow in porous media,, Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 54 (2003), 48. 
[18] 
B. Matkowsky and G. Sivashinsky, Propagation of a pulsating reaction front in solid fuel combustion,, SIAM J. Appl. Math., 35 (1978), 465. 
[19] 
J. Mota and S. Schecter, Combustion fronts in a porous medium with two layers,, Journal of Dynamics and Differential Equations, 18 (2006), 615. 
[20] 
J. Norbury and A. Stuart, Travelling combustion waves in a porous medium. part iexistence,, SIAM J. on Appl. Math., 48 (1988), 155. 
[21] 
J. Norbury and A. Stuart, Travelling combustion waves in a porous medium. part iistability,, SIAM J. on Appl. Math., 48 (1988), 374. 
[22] 
S. Schecter and D. Marchesin, Geometric singular perturbation analysis of oxidation heat pulses for twophase flow in porous media. dedicated to constantine dafermos on his 60th birthday,, Bulletin of the Brazilian Mathematical Society, 32 (2001), 237. 
[23] 
D. Schult, B. Matkowsky, V. Volpert and A. FernandezPello, Forced forward smolder combustion,, Combustion and Flame, 104 (1996), 1. 
[24] 
R. Seydel, Practical bifurcation and stability analysis,, Springer, (2010). 
[25] 
R. Weber, G. Mercer, H. Sidhu and B. Gray, Combustion waves for gases ($Le= 1$) and solids ($Le\to\infty$),, Proceedings of the Royal Society of London. Series A: Mathematical, 453 (1997), 1105. 
[26] 
Y. B. Zeldovich, G. I. Barenblatt, V. B. Librovich and G. M. Makhviladze, The mathematical theory of combustion and explosion,, Consultants Bureau, (1985). 
show all references
References:
[1] 
I. Akkutlu and Y. Yortsos, The dynamics of insitu combustion fronts in porous media,, J. of Combustion and Flame, 134 (2003), 229. 
[2] 
A. Aldushin and S. Kasparyan, Stability of stationary filtrational combustion waves,, Combustion, 17 (1981), 615. 
[3] 
A. Aldushin, I. Rumanov and B. Matkowsky, Maximal energy accumulation in a superadiabatic filtration combustion wave,, J. of Combustion and Flame, 118 (1999), 76. 
[4] 
A. Bayliss, G. Leaf and B. Matkowsky, Pulsating and chaotic dynamics near the extinction limit,, Combustion Science and Technology, 84 (1992), 253. 
[5] 
A. Bayliss and B. Matkowsky, Two routes to chaos in condensed phase combustion,, SIAM J. Appl. Math., 50 (1990), 437. 
[6] 
A. Bayliss and B. Matkowsky, From traveling waves to chaos in combustion,, SIAM Journal on Applied Mathematics, 54 (1994), 147. 
[7] 
I. Brailovsky and G. Sivashinsky, Chaotic dynamics in solid fuel combustion,, Physica D: Nonlinear Phenomena, 65 (1993), 191. 
[8] 
J. Bruining, A. Mailybaev and D. Marchesin, Filtration combustion in wet porous medium,, SIAM Journal on Applied Mathematics, 70 (2009), 1157. 
[9] 
G. Chapiro, L. Furtado, D. Marchesin and S. Schecter, Numerical analysis of combustion waves and riemann solutions in light porous foam, 2014,, Preprint at http://preprint.impa.br/visualizar?id=5912., (). 
[10] 
G. Chapiro, A. A. Mailybaev, A. Souza, D. Marchesin and J. Bruining, Asymptotic approximation of longtime solution for lowtemperature filtration combustion,, Comput. Geosciences, 16 (2012), 799. 
[11] 
G. Chapiro, D. Marchesin and S. Schecter, Combustion waves and Riemann solutions in light porous foam,, Journal of Hyperbolic Differential Equations, 11 (2014), 295. 
[12] 
M. Decker and D. Schult, Dynamics of smoulder waves near extinction,, Combustion Theory and Modelling, 8 (2004), 491. 
[13] 
A. Ghazaryan, Y. Latushkin, S. Schecter and A. de Souza, Stability of gasless combustion fronts in onedimensional solids,, Archive for Rational Mechanics and Analysis, 198 (2010), 981. 
[14] 
L. W. Lake, Enhanced oil recovery,, Old Tappan, (1989). 
[15] 
A. Mailybaev, J. Bruining and D. Marchesin, Analysis of in situ combustion of oil with pyrolysis and vaporization,, Combustion and Flame, 158 (2011), 1097. 
[16] 
A. Mailybaev, D. Marchesin and J. Bruining, Resonance in lowtemperature oxidation waves for porous media,, SIAM Journal on Mathematical Analysis, 43 (2011), 2230. 
[17] 
D. Marchesin and S. Schecter, Oxidation heat pulses in twophase expansive flow in porous media,, Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 54 (2003), 48. 
[18] 
B. Matkowsky and G. Sivashinsky, Propagation of a pulsating reaction front in solid fuel combustion,, SIAM J. Appl. Math., 35 (1978), 465. 
[19] 
J. Mota and S. Schecter, Combustion fronts in a porous medium with two layers,, Journal of Dynamics and Differential Equations, 18 (2006), 615. 
[20] 
J. Norbury and A. Stuart, Travelling combustion waves in a porous medium. part iexistence,, SIAM J. on Appl. Math., 48 (1988), 155. 
[21] 
J. Norbury and A. Stuart, Travelling combustion waves in a porous medium. part iistability,, SIAM J. on Appl. Math., 48 (1988), 374. 
[22] 
S. Schecter and D. Marchesin, Geometric singular perturbation analysis of oxidation heat pulses for twophase flow in porous media. dedicated to constantine dafermos on his 60th birthday,, Bulletin of the Brazilian Mathematical Society, 32 (2001), 237. 
[23] 
D. Schult, B. Matkowsky, V. Volpert and A. FernandezPello, Forced forward smolder combustion,, Combustion and Flame, 104 (1996), 1. 
[24] 
R. Seydel, Practical bifurcation and stability analysis,, Springer, (2010). 
[25] 
R. Weber, G. Mercer, H. Sidhu and B. Gray, Combustion waves for gases ($Le= 1$) and solids ($Le\to\infty$),, Proceedings of the Royal Society of London. Series A: Mathematical, 453 (1997), 1105. 
[26] 
Y. B. Zeldovich, G. I. Barenblatt, V. B. Librovich and G. M. Makhviladze, The mathematical theory of combustion and explosion,, Consultants Bureau, (1985). 
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