Spiral rotating waves of a geodesic curvature flow on the unit sphere

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May 2012, 17(3): 933-942. doi: 10.3934/dcdsb.2012.17.933

Spiral rotating waves of a geodesic curvature flow on the unit sphere

Bendong Lou ^1,

1.	Department of Mathematics, Tongji University, Shanghai 200092

Received December 2010 Revised July 2011 Published January 2012

Abstract
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This paper is concerned with a geodesic curvature flow on the unit sphere. In each zone between the equator and the circle with latitude $\theta_0 \in (0, \frac{\pi}{2} ]$, we give the existence and uniqueness of a spiral rotating wave of the geodesic curvature flow.

Keywords: Geodesic curvature flow, Yin-Yang curve., spiral rotating wave.

Mathematics Subject Classification: Primary: 53D25, 58J35; Secondary: 34B05, 35K5.

Citation: Bendong Lou. Spiral rotating waves of a geodesic curvature flow on the unit sphere. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 933-942. doi: 10.3934/dcdsb.2012.17.933

References:

[1]	M. Alfaro, D. Hilhorst and H. Matano, The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system, J. Differential Equations, 245 (2008), 505-565. doi: 10.1016/j.jde.2008.01.014.
[2]	S. J. Altschuler, Singularities of the curve shrinking flow for space curves, J. Differential Geom., 34 (1991), 491-514.
[3]	F. Amdjadi and J. Gomatam, Spiral waves on static and moving spherical domains, J. Comput. Appl. Math., 182 (2005), 472-486. doi: 10.1016/j.cam.2004.12.027.
[4]	X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141. doi: 10.1016/0022-0396(92)90146-E.
[5]	K.-S. Chou and X.-P. Zhu, "The Curve Shorting Problem," Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420035704.
[6]	P. C. Fife, "Dynamics of Internal Layers and Diffusive Interfaces," CBMS-NSF Regional Conference Series in Applied Mathematics, 53, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988.
[7]	J. Gomatam and F. Amdjadi, Reaction-diffusion equations on a sphere: Meandering of spiral waves, Physical Review E (3), 56 (1997), 3913-3919.
[8]	R. Ikota, N. Ishimura and T. Yamaguchi, On the structure of steady solutions for the kinematic model of spiral waves in excitable media, Japan J. Indust. Appl. Math., 15 (1998), 317-330. doi: 10.1007/BF03167407.
[9]	J. P. Keener, The core of the spiral, SIAM J. Appl. Math., 52 (1992), 1370-1390. doi: 10.1137/0152079.
[10]	J. P. Keener and J. J. Tyson, Spiral waves in the Belousov-Zhabotinski reaction, Phys. D, 21 (1986), 307-324. doi: 10.1016/0167-2789(86)90007-2.
[11]	B. Lou, Periodic rotating waves of a geodesic curvature flow on the sphere, Commun. Partial Differential Equations, 32 (2007), 525-541. doi: 10.1080/03605300701249663.
[12]	B. D. Lou and L. Zhou, Singular limit of FitzHugh-Nagumo equations on a sphere, ZAMM Z. Angew. Math. Mech., 88 (2008), 644-649. doi: 10.1002/zamm.200700144.
[13]	H. Matano, K.-I. Nakamura and B. Lou, Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit, Netw. Heterog. Media, 1 (2006), 537-568. doi: 10.3934/nhm.2006.1.537.
[14]	K.-I. Nakamura, H. Matano, D. Hilhorst and R. Schätzle, Singular limit of a reaction-diffusion equation with a spatially inhomogeneous reaction term, J. Statist. Phys., 95 (1999), 1165-1185. doi: 10.1023/A:1004518904533.
[15]	T. Ogiwara and K.-I. Nakamura, Spiral traveling wave solutions of nonlinear diffusion equations related to a model of spiral crystal growth, Publ. Res. Inst. Math. Sci., 39 (2003), 767-783. doi: 10.2977/prims/1145476046.

show all references

References:

[1]	M. Alfaro, D. Hilhorst and H. Matano, The singular limit of the Allen-Cahn equation and the FitzHugh-Nagumo system, J. Differential Equations, 245 (2008), 505-565. doi: 10.1016/j.jde.2008.01.014.
[2]	S. J. Altschuler, Singularities of the curve shrinking flow for space curves, J. Differential Geom., 34 (1991), 491-514.
[3]	F. Amdjadi and J. Gomatam, Spiral waves on static and moving spherical domains, J. Comput. Appl. Math., 182 (2005), 472-486. doi: 10.1016/j.cam.2004.12.027.
[4]	X. Chen, Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations, 96 (1992), 116-141. doi: 10.1016/0022-0396(92)90146-E.
[5]	K.-S. Chou and X.-P. Zhu, "The Curve Shorting Problem," Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420035704.
[6]	P. C. Fife, "Dynamics of Internal Layers and Diffusive Interfaces," CBMS-NSF Regional Conference Series in Applied Mathematics, 53, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988.
[7]	J. Gomatam and F. Amdjadi, Reaction-diffusion equations on a sphere: Meandering of spiral waves, Physical Review E (3), 56 (1997), 3913-3919.
[8]	R. Ikota, N. Ishimura and T. Yamaguchi, On the structure of steady solutions for the kinematic model of spiral waves in excitable media, Japan J. Indust. Appl. Math., 15 (1998), 317-330. doi: 10.1007/BF03167407.
[9]	J. P. Keener, The core of the spiral, SIAM J. Appl. Math., 52 (1992), 1370-1390. doi: 10.1137/0152079.
[10]	J. P. Keener and J. J. Tyson, Spiral waves in the Belousov-Zhabotinski reaction, Phys. D, 21 (1986), 307-324. doi: 10.1016/0167-2789(86)90007-2.
[11]	B. Lou, Periodic rotating waves of a geodesic curvature flow on the sphere, Commun. Partial Differential Equations, 32 (2007), 525-541. doi: 10.1080/03605300701249663.
[12]	B. D. Lou and L. Zhou, Singular limit of FitzHugh-Nagumo equations on a sphere, ZAMM Z. Angew. Math. Mech., 88 (2008), 644-649. doi: 10.1002/zamm.200700144.
[13]	H. Matano, K.-I. Nakamura and B. Lou, Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit, Netw. Heterog. Media, 1 (2006), 537-568. doi: 10.3934/nhm.2006.1.537.
[14]	K.-I. Nakamura, H. Matano, D. Hilhorst and R. Schätzle, Singular limit of a reaction-diffusion equation with a spatially inhomogeneous reaction term, J. Statist. Phys., 95 (1999), 1165-1185. doi: 10.1023/A:1004518904533.
[15]	T. Ogiwara and K.-I. Nakamura, Spiral traveling wave solutions of nonlinear diffusion equations related to a model of spiral crystal growth, Publ. Res. Inst. Math. Sci., 39 (2003), 767-783. doi: 10.2977/prims/1145476046.

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