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Traveling wave solutions of a 3-component reaction-diffusion model in smoldering combustion
Heterogeneity-induced spot dynamics for a three-component reaction-diffusion system
1. | Research Institute for Electronic Science, Hokkaido University, Sapporo 060-0813, Japan |
2. | Faculty of Photonics Science, Chitose Institute of Science and Technology, Chitose 066-8655, Japan |
3. | Northeast Institute of Geography and Agroecology, Chinese Academy of Sciences, Harbin 150081, China |
References:
[1] |
M. Bär, M. Eiswirth, H.-H. Rotermund and G. Ertl, Solitary-wave phenomena in an excitable surface reaction, Phys. Rev. Lett., 69 (1992), 945-948.
doi: 10.1103/PhysRevLett.69.945. |
[2] |
M. Bär, E. Meron and C. Utzny, Pattern formation on anisotropic and heterogeneous catalytic surfaces, Chaos, 12 (2002), 204-214.
doi: 10.1063/1.1450565. |
[3] |
M. Bode, Front-bifurcations in reaction-diffusion systems with inhomogeneous parameter distributions, Physica D, 106 (1997), 270-286.
doi: 10.1016/S0167-2789(97)00050-X. |
[4] |
M. Bode, A. W. Liehr, C. P. Schenk and H.-G.Purwins, Interaction of dissipative solitons: particle-like behaviour of localized structures in a threecomponent reaction-diffusion system, Physica D, 161 (2002), 45-66.
doi: 10.1016/S0167-2789(01)00360-8. |
[5] |
P. C. Bressloff, S. E. Folias, A. Prat and Y.-X. Li, Oscillatory waves in inhomogeneous neural media, Phys. Rev. Lett., 91 (2003), 178101.
doi: 10.1103/PhysRevLett.91.178101. |
[6] |
E. J. Doedel, R. C. Paffenroth, A. R. Champneys, T. F. Fairgrieve, Yu, B. Sandstede and X. Wang, AUTO 2000: Continuation and bifurcation software for ordinary differential equations (with homcont), Technical report, California Institute of Technology, 2001. |
[7] |
A. Doelman, P. van Heijster and T. Kaper, Pulse dynamics in a three-component system: existence analysys, J. Dyn. Diff. Equat., 21 (2009), 73-115.
doi: 10.1007/s10884-008-9125-2. |
[8] |
S.-I. Ei, M. Mumura and Nagayama, Interacting spots in reaction diffusion systems, Discrete Contin. Dyn. Syst., 14 (2006), 31-62. |
[9] |
G. B. Ermentrout and J. Rinzel, Reflected waves in an inhomogeneous excitable medium, SIAM J. Appl. Math., 56 (1996), 1107-1128.
doi: 10.1137/S0036139994276793. |
[10] |
M. Gutman, I. Aviram and A. Rabinovitch, Pseudoreflection from interface between two oscillatory media: Extended driver, Phys. Rev. E, 69 (2004), 016211.
doi: 10.1103/PhysRevE.69.016211. |
[11] |
A. Hagberg, E. Meron, I. Rubinstein and B. Zaltzman, Controlling domain patterns far from equilibrium, Phy. Rev. Lett., 76 (1996), 427-430.
doi: 10.1103/PhysRevLett.76.427. |
[12] |
A. Hagberg, E. Meron, I. Rubinstein and B. Zaltzman, Order parameter equations for front bifurcations: planar and circular fronts, Phy. Rev. E, 55 (1997), 4450-4457.
doi: 10.1103/PhysRevE.55.4450. |
[13] |
Y. Hayase and T Ohta, Self-replicating pulses and sierpinski gaskets in excitable media, Phy. Rev. E, 62 (2000), 5998-6003.
doi: 10.1103/PhysRevE.62.5998. |
[14] |
P. van Heijster and B. Sandstede, Planar radial spots in a three-component FitzHugh-Nagumo system, J. Nonlinear Sci., 21 (2011), in press. |
[15] |
H. Ikeda and M. Mimura, Wave-blocking phenomena in bistable reaction-diffusion systems, SIAM J. Appl. Math., 49 (1989) 515-538
doi: 10.1103/PhysRevE.62.5998. |
[16] |
J. Keener and J. Sneyd, "Mathematical Physiology," Springer-Verlag, New York, 1998. |
[17] |
R. B. Lehoucq, D. C. Sorensen and C. Yang, "ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods," SIAM, 1998. |
[18] |
T. J. Lewis and J. P. Keener, Wave-block in excitable media due to regions of depressed excitability, SIAM J. Appl. Math., 61 (2000), 293-316.
doi: 10.1137/S0036139998349298. |
[19] |
H. Meinhardt, "The Algorithmmic Beauty of Sea Shells," PSpringer-Verlag, Heidelberg, 1995. |
[20] |
J. Miyazaki and S. Kinoshita, Stopping and initiation of a chemical pulse at the interface of excitable media with different diffusivity, Phys. Rev. E, 76 (2007), 066201.
doi: 10.1103/PhysRevE.76.066201. |
[21] |
Y. Nishiura, T. Teramoto and K.-I. Ueda, Dynamic transitions through scattors in dissipative systems, Chaos, 13 (2003), 962-972.
doi: 10.1063/1.1592131. |
[22] |
Y. Nishiura, T. Teramoto and K.-I. Ueda, Scattering and separators in dissipative systems, Phys. Rev. E, 67 (2003), 056210.
doi: 10.1103/PhysRevE.67.056210. |
[23] |
Y. Nishiura, T. Teramoto, X. Yuan and K.-I. Ueda, Dynamics of traveling pulses in heterogenous media, Chaos, 17 (2007), 037104.
doi: 10.1063/1.2778553. |
[24] |
Y. Nishiura, T. Teramoto and K.-I. Ueda, Scattering of traveling spots in dissipative systems, Chaos, 15 (2005), 047509.
doi: 10.1063/1.2087127. |
[25] |
J. P. Pauwelussen, Nerve impulse propagation in a branching nerve system: a simple model, Physica D, 4 (1981), 67-88.
doi: 10.1016/0167-2789(81)90005-1. |
[26] |
A. M. Pertsov, E. A. Ermakova and E. E. Shnol, On the diffraction of autowaves, Physica D, 44 (1990), 178-190.
doi: 10.1016/0167-2789(90)90054-S. |
[27] |
A. Prat and Y.-X. Li and P. Bressloff, Inhomogeneity-induced bifurcation of stationary and oscillatory pulses, Physica D, 202 (2005), 177-199.
doi: 10.1016/j.physd.2005.02.005. |
[28] |
H.-G. Purwins, H. U. Bödeker and Sh. Amiranashvili, Dissipative solitons, Advances in Physics, 59 (2010), 485-701.
doi: 10.1080/00018732.2010.498228. |
[29] |
A. G. Salinger, N. M. Bou-Rabee, E. A. Burroughs, R. B. Lehoucq, R. P. Pawlowski, L. A. Romero and E. D. Wilkes, "LOCA 1.1: Library of Continuation Algorithms, Theory and Implementation Manual," Sandia National Laboratories, Albuquerque, USA, 2002.
doi: 10.2172/800778. |
[30] |
C. P. Schenk, M. Or-Guil, M. Bode and H.-G. Purwins, Interacting pulses in three-component reaction-diffusion systems on two-dimensional domains, Phys. Rev. Lett., 78 (1997), 3781-3784.
doi: 10.1103/PhysRevLett.78.3781. |
[31] |
P. Schütz, M. Bode and H. G. Purwins, Bifurcations of front dynamics in a reaction-diffusion system with spatial inhomogeneities, Physica D, 82 (1995), 270-286. |
[32] |
R. Seydel, "Practical Bifurcation and Stability Analysis," Springer-Verlag, 1994. |
[33] |
A. J. Steele, M. Tinsley and K. Showalter, Collective behavior of stabilized reaction-diffusion waves, Chaos, 18 (2008), 026108.
doi: 10.1063/1.2900386. |
[34] |
T. Teramoto, K. Suzuki and Y. Nishiura, Rotational motion of traveling spots in dissipative systems, Phys. Rev. E, 80 (2009), 046208.
doi: 10.1103/PhysRevE.80.046208. |
[35] |
T. Teramoto, X. Yuan, M. Bär and Y. Nishiura, Onset of inidirectional pulse propagation in an excitable medium with asymmetric heterogeneity, Phys. Rev. E, 79 (2009), 046205.
doi: 10.1103/PhysRevE.79.046205. |
[36] |
T. Teramoto, Traveling spots through a line of heterogeneity, unpublished. |
[37] |
T. Teramoto, K.-I. Ueda and Y. Nishiura, Phase-dependent output of scattering process for traveling breathers, Phys. Rev. E, 69 (2004), 056224.
doi: 10.1103/PhysRevE.69.056224. |
[38] |
M. R. Tinsley, A. J. Steele and K. Showalter, Collective behavior of particle-like chemical waves, Eur. Phys. J. Special Topics, 165 (2008), 161-167.
doi: 10.1140/epjst/e2008-00859-7. |
[39] |
R. S. Tuminaro, M. Heroux, S. A. Hutchinson and J. N. Shadid, "Official Aztec User's Guide: Version 2.1," Technical Report, SAND99-8801J, December, 1999. |
[40] |
V. K. Vanag and I. R. Epstein, Localized patterns in reaction-diffusion systems, Chaos, 17 (2007), 037110.
doi: 10.1063/1.2752494. |
[41] |
J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.
doi: 10.1137/S0036144599364296. |
[42] |
X. Yuan, T. Teramoto and Y. Nishiura, Heterogeneity-induced defect bifurcation and pulse dynamics for a three-component reaction-diffusion system, Phys. Rev. E, 75 (2007), 036220.
doi: 10.1103/PhysRevE.75.036220. |
[43] |
A. M. Zhabotinsky, M. D. Eager and I. R. Epstein, Refraction and reflection of chemical waves, Phys. Rev. Lett., 71 (1993), 1526-1529.
doi: 10.1103/PhysRevLett.71.1526. |
show all references
References:
[1] |
M. Bär, M. Eiswirth, H.-H. Rotermund and G. Ertl, Solitary-wave phenomena in an excitable surface reaction, Phys. Rev. Lett., 69 (1992), 945-948.
doi: 10.1103/PhysRevLett.69.945. |
[2] |
M. Bär, E. Meron and C. Utzny, Pattern formation on anisotropic and heterogeneous catalytic surfaces, Chaos, 12 (2002), 204-214.
doi: 10.1063/1.1450565. |
[3] |
M. Bode, Front-bifurcations in reaction-diffusion systems with inhomogeneous parameter distributions, Physica D, 106 (1997), 270-286.
doi: 10.1016/S0167-2789(97)00050-X. |
[4] |
M. Bode, A. W. Liehr, C. P. Schenk and H.-G.Purwins, Interaction of dissipative solitons: particle-like behaviour of localized structures in a threecomponent reaction-diffusion system, Physica D, 161 (2002), 45-66.
doi: 10.1016/S0167-2789(01)00360-8. |
[5] |
P. C. Bressloff, S. E. Folias, A. Prat and Y.-X. Li, Oscillatory waves in inhomogeneous neural media, Phys. Rev. Lett., 91 (2003), 178101.
doi: 10.1103/PhysRevLett.91.178101. |
[6] |
E. J. Doedel, R. C. Paffenroth, A. R. Champneys, T. F. Fairgrieve, Yu, B. Sandstede and X. Wang, AUTO 2000: Continuation and bifurcation software for ordinary differential equations (with homcont), Technical report, California Institute of Technology, 2001. |
[7] |
A. Doelman, P. van Heijster and T. Kaper, Pulse dynamics in a three-component system: existence analysys, J. Dyn. Diff. Equat., 21 (2009), 73-115.
doi: 10.1007/s10884-008-9125-2. |
[8] |
S.-I. Ei, M. Mumura and Nagayama, Interacting spots in reaction diffusion systems, Discrete Contin. Dyn. Syst., 14 (2006), 31-62. |
[9] |
G. B. Ermentrout and J. Rinzel, Reflected waves in an inhomogeneous excitable medium, SIAM J. Appl. Math., 56 (1996), 1107-1128.
doi: 10.1137/S0036139994276793. |
[10] |
M. Gutman, I. Aviram and A. Rabinovitch, Pseudoreflection from interface between two oscillatory media: Extended driver, Phys. Rev. E, 69 (2004), 016211.
doi: 10.1103/PhysRevE.69.016211. |
[11] |
A. Hagberg, E. Meron, I. Rubinstein and B. Zaltzman, Controlling domain patterns far from equilibrium, Phy. Rev. Lett., 76 (1996), 427-430.
doi: 10.1103/PhysRevLett.76.427. |
[12] |
A. Hagberg, E. Meron, I. Rubinstein and B. Zaltzman, Order parameter equations for front bifurcations: planar and circular fronts, Phy. Rev. E, 55 (1997), 4450-4457.
doi: 10.1103/PhysRevE.55.4450. |
[13] |
Y. Hayase and T Ohta, Self-replicating pulses and sierpinski gaskets in excitable media, Phy. Rev. E, 62 (2000), 5998-6003.
doi: 10.1103/PhysRevE.62.5998. |
[14] |
P. van Heijster and B. Sandstede, Planar radial spots in a three-component FitzHugh-Nagumo system, J. Nonlinear Sci., 21 (2011), in press. |
[15] |
H. Ikeda and M. Mimura, Wave-blocking phenomena in bistable reaction-diffusion systems, SIAM J. Appl. Math., 49 (1989) 515-538
doi: 10.1103/PhysRevE.62.5998. |
[16] |
J. Keener and J. Sneyd, "Mathematical Physiology," Springer-Verlag, New York, 1998. |
[17] |
R. B. Lehoucq, D. C. Sorensen and C. Yang, "ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods," SIAM, 1998. |
[18] |
T. J. Lewis and J. P. Keener, Wave-block in excitable media due to regions of depressed excitability, SIAM J. Appl. Math., 61 (2000), 293-316.
doi: 10.1137/S0036139998349298. |
[19] |
H. Meinhardt, "The Algorithmmic Beauty of Sea Shells," PSpringer-Verlag, Heidelberg, 1995. |
[20] |
J. Miyazaki and S. Kinoshita, Stopping and initiation of a chemical pulse at the interface of excitable media with different diffusivity, Phys. Rev. E, 76 (2007), 066201.
doi: 10.1103/PhysRevE.76.066201. |
[21] |
Y. Nishiura, T. Teramoto and K.-I. Ueda, Dynamic transitions through scattors in dissipative systems, Chaos, 13 (2003), 962-972.
doi: 10.1063/1.1592131. |
[22] |
Y. Nishiura, T. Teramoto and K.-I. Ueda, Scattering and separators in dissipative systems, Phys. Rev. E, 67 (2003), 056210.
doi: 10.1103/PhysRevE.67.056210. |
[23] |
Y. Nishiura, T. Teramoto, X. Yuan and K.-I. Ueda, Dynamics of traveling pulses in heterogenous media, Chaos, 17 (2007), 037104.
doi: 10.1063/1.2778553. |
[24] |
Y. Nishiura, T. Teramoto and K.-I. Ueda, Scattering of traveling spots in dissipative systems, Chaos, 15 (2005), 047509.
doi: 10.1063/1.2087127. |
[25] |
J. P. Pauwelussen, Nerve impulse propagation in a branching nerve system: a simple model, Physica D, 4 (1981), 67-88.
doi: 10.1016/0167-2789(81)90005-1. |
[26] |
A. M. Pertsov, E. A. Ermakova and E. E. Shnol, On the diffraction of autowaves, Physica D, 44 (1990), 178-190.
doi: 10.1016/0167-2789(90)90054-S. |
[27] |
A. Prat and Y.-X. Li and P. Bressloff, Inhomogeneity-induced bifurcation of stationary and oscillatory pulses, Physica D, 202 (2005), 177-199.
doi: 10.1016/j.physd.2005.02.005. |
[28] |
H.-G. Purwins, H. U. Bödeker and Sh. Amiranashvili, Dissipative solitons, Advances in Physics, 59 (2010), 485-701.
doi: 10.1080/00018732.2010.498228. |
[29] |
A. G. Salinger, N. M. Bou-Rabee, E. A. Burroughs, R. B. Lehoucq, R. P. Pawlowski, L. A. Romero and E. D. Wilkes, "LOCA 1.1: Library of Continuation Algorithms, Theory and Implementation Manual," Sandia National Laboratories, Albuquerque, USA, 2002.
doi: 10.2172/800778. |
[30] |
C. P. Schenk, M. Or-Guil, M. Bode and H.-G. Purwins, Interacting pulses in three-component reaction-diffusion systems on two-dimensional domains, Phys. Rev. Lett., 78 (1997), 3781-3784.
doi: 10.1103/PhysRevLett.78.3781. |
[31] |
P. Schütz, M. Bode and H. G. Purwins, Bifurcations of front dynamics in a reaction-diffusion system with spatial inhomogeneities, Physica D, 82 (1995), 270-286. |
[32] |
R. Seydel, "Practical Bifurcation and Stability Analysis," Springer-Verlag, 1994. |
[33] |
A. J. Steele, M. Tinsley and K. Showalter, Collective behavior of stabilized reaction-diffusion waves, Chaos, 18 (2008), 026108.
doi: 10.1063/1.2900386. |
[34] |
T. Teramoto, K. Suzuki and Y. Nishiura, Rotational motion of traveling spots in dissipative systems, Phys. Rev. E, 80 (2009), 046208.
doi: 10.1103/PhysRevE.80.046208. |
[35] |
T. Teramoto, X. Yuan, M. Bär and Y. Nishiura, Onset of inidirectional pulse propagation in an excitable medium with asymmetric heterogeneity, Phys. Rev. E, 79 (2009), 046205.
doi: 10.1103/PhysRevE.79.046205. |
[36] |
T. Teramoto, Traveling spots through a line of heterogeneity, unpublished. |
[37] |
T. Teramoto, K.-I. Ueda and Y. Nishiura, Phase-dependent output of scattering process for traveling breathers, Phys. Rev. E, 69 (2004), 056224.
doi: 10.1103/PhysRevE.69.056224. |
[38] |
M. R. Tinsley, A. J. Steele and K. Showalter, Collective behavior of particle-like chemical waves, Eur. Phys. J. Special Topics, 165 (2008), 161-167.
doi: 10.1140/epjst/e2008-00859-7. |
[39] |
R. S. Tuminaro, M. Heroux, S. A. Hutchinson and J. N. Shadid, "Official Aztec User's Guide: Version 2.1," Technical Report, SAND99-8801J, December, 1999. |
[40] |
V. K. Vanag and I. R. Epstein, Localized patterns in reaction-diffusion systems, Chaos, 17 (2007), 037110.
doi: 10.1063/1.2752494. |
[41] |
J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230.
doi: 10.1137/S0036144599364296. |
[42] |
X. Yuan, T. Teramoto and Y. Nishiura, Heterogeneity-induced defect bifurcation and pulse dynamics for a three-component reaction-diffusion system, Phys. Rev. E, 75 (2007), 036220.
doi: 10.1103/PhysRevE.75.036220. |
[43] |
A. M. Zhabotinsky, M. D. Eager and I. R. Epstein, Refraction and reflection of chemical waves, Phys. Rev. Lett., 71 (1993), 1526-1529.
doi: 10.1103/PhysRevLett.71.1526. |
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