American Institute of Mathematical Sciences

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March  2013, 12(2): 1049-1063. doi: 10.3934/cpaa.2013.12.1049

Existence of a rotating wave pattern in a disk for a wave front interaction model

Jong-Shenq Guo 1, , Hirokazu Ninomiya 2, and  Chin-Chin Wu 3,

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

Received  July 2011 Revised  November 2011 Published  September 2012

We study the rotating wave patterns in an excitable medium in a disk. This wave pattern is rotating along the given disk boundary with a constant angular speed. To study this pattern we use the wave front interaction model proposed by Zykov in 2007. This model is derived from the FitzHugh-Nagumo equation and it can be described by two systems of ordinary differential equations for wave front and wave back respectively. Using a delicate shooting argument with the help of the comparison principle, we derive the existence and uniqueness of rotating wave patterns for any admissible angular speed with convex front in a given disk.
Keywords: Rotating wave pattern, back, front, angular speed..
Mathematics Subject Classification: Primary: 34B15, 34B60; Secondary: 35K5.
Citation: Jong-Shenq Guo, Hirokazu Ninomiya, Chin-Chin Wu. Existence of a rotating wave pattern in a disk for a wave front interaction model. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1049-1063. doi: 10.3934/cpaa.2013.12.1049
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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