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The explicit nonlinear wave solutions of the generalized $b$-equation
Existence of a rotating wave pattern in a disk for a wave front interaction model
1. | Department of Mathematics, Tamkang University, 151, Ying-Chuan Road, Tamsui, Taipei County 25137 |
2. | Department of Mathematics, Meiji University, 1-1-1 Higashi-mita, Tama-ku, Kawasaki, 214-8571 |
3. | Department of Applied Mathematics, National Chung Hsing University, 250, Kuo Kuang Road, Taichung 402 |
References:
[1] |
B. Fiedler and A. Scheel, Spatio-temporal dynamics of reaction-diffusion patterns, In "Trends in Nonlinear Analysis" (M. Kirkilionis, S. Kromker, R. Rannacher and F. Tomi F. eds.), 23-152, Berlin, Heidelberg, New York, Springer, 2003. |
[2] |
P. bC. Fife, Understanding the patterns in the BZ reagent, J. Statist. Phys., 39 (1985), 687-703.
doi: 10.1007/BF01008360. |
[3] |
J.-S. Guo, K.-I. Nakamura, T. Ogiwara and J.-C. Tsai, On the steadily rotating spirals, Japan J. Indust. Appl. Math., 23 (2006), 1-19.
doi: 10.1007/BF03167495. |
[4] |
J.-S. Guo, H. Ninomiya and J.-C. Tsai, Existence and uniqueness of stabilized propagating wave segments in wave front interaction model, Physica D: Nonlinear Phenomena, 239 (2010), 230-239.
doi: 10.1016/j.physd.2009.11.001. |
[5] |
P. Hartman, "Ordinary Differential Equations," SIAM, Philadelphia, 2002.
doi: 10.1137/1.9780898719222. |
[6] |
A. Karma, Universal limit of spiral wave propagation in excitable media, Phys. Review Letters, 66 (1991), 2274-2277.
doi: 10.1103/PhysRevLett.66.2274. |
[7] |
J. P. Keener and J. J. Tyson, Spiral waves in the Belousov-Zhabotinskii reaction, Physical D, 21 (1986), 307-324.
doi: 10.1016/0167-2789(86)90007-2. |
[8] |
W. F. Loomis, "The Development of Dictyostelium Discoideum," Academic Press, New York, 1982. |
[9] |
E. Meron, Pattern formation in excitable media, Phys. Rep., 218 (1992), 1-66.
doi: 10.1016/0370-1573(92)90098-K. |
[10] |
E. Mihaliuk, T. Sakurai, F. Chirila and K. Showalter, Experimental and theoretical studies of feedback stabilization of propagating wave segments, Faraday Discussions, 120 (2002), 383-394.
doi: 10.1039/B103431F. |
[11] |
E. Mihaliuk, T. Sakurai, F. Chirila and K. Showalter, Feedback stabilization of unstable propagating waves, Phys. Review E. 65 (2002), 065602.
doi: 10.1103/PhysRevE.65.065602. |
[12] |
A. S. Mikhailov, Modeling pattern formation in excitable media: The Legacy of Norbert Wiener, In "Epilepsy as a Dynamic Disease" (J. Milton and P. Jung eds.), Berlin, Heidelberg, New York, Springer, 2003. |
[13] |
A. S. Mikhailov and V. S. Zykov, Kinematical theory of spiral waves in excitable media: comparison with numerical simulations, Physica D, 52 (1991), 379-397.
doi: 10.1016/0167-2789(91)90134-U. |
[14] |
J. D. Murray, "Mathematical Biology. I: An introduction," Springer-Verlag, New York, 2004. |
[15] |
P. Pelce and J. Sun, On the stability of steadily rotating waves in the free boundary formulation, Physica D, 63 (1993), 273-281.
doi: 10.1016/0167-2789(93)90111-D. |
[16] |
Á. Tóth, V. Gaspar and K. Showalter, Signal transmission in chemical systems: propagation of chemical waves through capillary tubes, J. Phys. Chem., 98 (1994), 522-531.
doi: 10.1021/j100053a029. |
[17] |
J. J. Tyson and J. P. Keener, Singular perturbation theory of traveling waves in excitable media (a review), Physica D, 32 (1988), 327-361.
doi: 10.1016/0167-2789(88)90062-0. |
[18] |
N. Wiener and A. Rosenblueth, The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle, Arch. Inst. Cardiol. Mexico, 16 (1946), 205-265. |
[19] |
W. F. Winfree, "When Time Breaks Down," Princeton Univ. Press, Princeton, 1987. |
[20] |
V. S. Zykov, "Simulation of Wave Process in Excitable Media," Manchester University Press, 1984. |
[21] |
V. S. Zykov and K. Showalter, Wave front interaction model of stabilized propagating wave segments, Phys. Review Letters, 94 (2005), 068302.
doi: 10.1103/PhysRevLett.94.068302. |
[22] |
V. S. Zykov, Selection mechanism for rotating patterns in weakly excitable media, Physical Review E, 75 (2007), 046203.
doi: 10.1103/PhysRevE.75.046203. |
show all references
References:
[1] |
B. Fiedler and A. Scheel, Spatio-temporal dynamics of reaction-diffusion patterns, In "Trends in Nonlinear Analysis" (M. Kirkilionis, S. Kromker, R. Rannacher and F. Tomi F. eds.), 23-152, Berlin, Heidelberg, New York, Springer, 2003. |
[2] |
P. bC. Fife, Understanding the patterns in the BZ reagent, J. Statist. Phys., 39 (1985), 687-703.
doi: 10.1007/BF01008360. |
[3] |
J.-S. Guo, K.-I. Nakamura, T. Ogiwara and J.-C. Tsai, On the steadily rotating spirals, Japan J. Indust. Appl. Math., 23 (2006), 1-19.
doi: 10.1007/BF03167495. |
[4] |
J.-S. Guo, H. Ninomiya and J.-C. Tsai, Existence and uniqueness of stabilized propagating wave segments in wave front interaction model, Physica D: Nonlinear Phenomena, 239 (2010), 230-239.
doi: 10.1016/j.physd.2009.11.001. |
[5] |
P. Hartman, "Ordinary Differential Equations," SIAM, Philadelphia, 2002.
doi: 10.1137/1.9780898719222. |
[6] |
A. Karma, Universal limit of spiral wave propagation in excitable media, Phys. Review Letters, 66 (1991), 2274-2277.
doi: 10.1103/PhysRevLett.66.2274. |
[7] |
J. P. Keener and J. J. Tyson, Spiral waves in the Belousov-Zhabotinskii reaction, Physical D, 21 (1986), 307-324.
doi: 10.1016/0167-2789(86)90007-2. |
[8] |
W. F. Loomis, "The Development of Dictyostelium Discoideum," Academic Press, New York, 1982. |
[9] |
E. Meron, Pattern formation in excitable media, Phys. Rep., 218 (1992), 1-66.
doi: 10.1016/0370-1573(92)90098-K. |
[10] |
E. Mihaliuk, T. Sakurai, F. Chirila and K. Showalter, Experimental and theoretical studies of feedback stabilization of propagating wave segments, Faraday Discussions, 120 (2002), 383-394.
doi: 10.1039/B103431F. |
[11] |
E. Mihaliuk, T. Sakurai, F. Chirila and K. Showalter, Feedback stabilization of unstable propagating waves, Phys. Review E. 65 (2002), 065602.
doi: 10.1103/PhysRevE.65.065602. |
[12] |
A. S. Mikhailov, Modeling pattern formation in excitable media: The Legacy of Norbert Wiener, In "Epilepsy as a Dynamic Disease" (J. Milton and P. Jung eds.), Berlin, Heidelberg, New York, Springer, 2003. |
[13] |
A. S. Mikhailov and V. S. Zykov, Kinematical theory of spiral waves in excitable media: comparison with numerical simulations, Physica D, 52 (1991), 379-397.
doi: 10.1016/0167-2789(91)90134-U. |
[14] |
J. D. Murray, "Mathematical Biology. I: An introduction," Springer-Verlag, New York, 2004. |
[15] |
P. Pelce and J. Sun, On the stability of steadily rotating waves in the free boundary formulation, Physica D, 63 (1993), 273-281.
doi: 10.1016/0167-2789(93)90111-D. |
[16] |
Á. Tóth, V. Gaspar and K. Showalter, Signal transmission in chemical systems: propagation of chemical waves through capillary tubes, J. Phys. Chem., 98 (1994), 522-531.
doi: 10.1021/j100053a029. |
[17] |
J. J. Tyson and J. P. Keener, Singular perturbation theory of traveling waves in excitable media (a review), Physica D, 32 (1988), 327-361.
doi: 10.1016/0167-2789(88)90062-0. |
[18] |
N. Wiener and A. Rosenblueth, The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle, Arch. Inst. Cardiol. Mexico, 16 (1946), 205-265. |
[19] |
W. F. Winfree, "When Time Breaks Down," Princeton Univ. Press, Princeton, 1987. |
[20] |
V. S. Zykov, "Simulation of Wave Process in Excitable Media," Manchester University Press, 1984. |
[21] |
V. S. Zykov and K. Showalter, Wave front interaction model of stabilized propagating wave segments, Phys. Review Letters, 94 (2005), 068302.
doi: 10.1103/PhysRevLett.94.068302. |
[22] |
V. S. Zykov, Selection mechanism for rotating patterns in weakly excitable media, Physical Review E, 75 (2007), 046203.
doi: 10.1103/PhysRevE.75.046203. |
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