American Institute of Mathematical Sciences

Advanced
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

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 & 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.  Google Scholar

[2]

S. J. Altschuler, Singularities of the curve shrinking flow for space curves, J. Differential Geom., 34 (1991), 491-514.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

K.-S. Chou and X.-P. Zhu, "The Curve Shorting Problem," Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420035704.  Google Scholar

[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.  Google Scholar

[7]

J. Gomatam and F. Amdjadi, Reaction-diffusion equations on a sphere: Meandering of spiral waves, Physical Review E (3), 56 (1997), 3913-3919.  Google Scholar

[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.  Google Scholar

[9]

J. P. Keener, The core of the spiral, SIAM J. Appl. Math., 52 (1992), 1370-1390. doi: 10.1137/0152079.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

S. J. Altschuler, Singularities of the curve shrinking flow for space curves, J. Differential Geom., 34 (1991), 491-514.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

K.-S. Chou and X.-P. Zhu, "The Curve Shorting Problem," Chapman & Hall/CRC, Boca Raton, FL, 2001. doi: 10.1201/9781420035704.  Google Scholar

[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.  Google Scholar

[7]

J. Gomatam and F. Amdjadi, Reaction-diffusion equations on a sphere: Meandering of spiral waves, Physical Review E (3), 56 (1997), 3913-3919.  Google Scholar

[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.  Google Scholar

[9]

J. P. Keener, The core of the spiral, SIAM J. Appl. Math., 52 (1992), 1370-1390. doi: 10.1137/0152079.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Tetsuya Ishiwata, Takeshi Ohtsuka. Evolution of a spiral-shaped polygonal curve by the crystalline curvature flow with a pinned tip. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5261-5295. doi: 10.3934/dcdsb.2019058
[2]

Miroslav KolÁŘ, Michal BeneŠ, Daniel ŠevČoviČ. Area preserving geodesic curvature driven flow of closed curves on a surface. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3671-3689. doi: 10.3934/dcdsb.2017148
[3]

Bendong Lou. Traveling wave solutions of a generalized curvature flow equation in the plane. Conference Publications, 2007, 2007 (Special) : 687-693. doi: 10.3934/proc.2007.2007.687
[4]

Tetsuya Ishiwata. On spiral solutions to generalized crystalline motion with a rotating tip motion. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 881-888. doi: 10.3934/dcdss.2015.8.881
[5]

Huaiyu Jian, Hongjie Ju, Wei Sun. Traveling fronts of curve flow with external force field. Communications on Pure & Applied Analysis, 2010, 9 (4) : 975-986. doi: 10.3934/cpaa.2010.9.975
[6]

Zhenqi Jenny Wang. The twisted cohomological equation over the geodesic flow. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 3923-3940. doi: 10.3934/dcds.2019158
[7]

Dieter Mayer, Fredrik Strömberg. Symbolic dynamics for the geodesic flow on Hecke surfaces. Journal of Modern Dynamics, 2008, 2 (4) : 581-627. doi: 10.3934/jmd.2008.2.581
[8]

Mark Pollicott. Closed geodesic distribution for manifolds of non-positive curvature. Discrete & Continuous Dynamical Systems, 1996, 2 (2) : 153-161. doi: 10.3934/dcds.1996.2.153
[9]

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 & Applied Analysis, 2013, 12 (2) : 1049-1063. doi: 10.3934/cpaa.2013.12.1049
[10]

Anke D. Pohl. Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orbifolds. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2173-2241. doi: 10.3934/dcds.2014.34.2173
[11]

Vladimir S. Matveev and Petar J. Topalov. Metric with ergodic geodesic flow is completely determined by unparameterized geodesics. Electronic Research Announcements, 2000, 6: 98-104.

[12]

Jonatan Lenells. Weak geodesic flow and global solutions of the Hunter-Saxton equation. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 643-656. doi: 10.3934/dcds.2007.18.643
[13]

Pablo D. Carrasco, Túlio Vales. A symmetric Random Walk defined by the time-one map of a geodesic flow. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2891-2905. doi: 10.3934/dcds.2020390
[14]

Rafael O. Ruggiero. Shadowing of geodesics, weak stability of the geodesic flow and global hyperbolic geometry. Discrete & Continuous Dynamical Systems, 2006, 14 (2) : 365-383. doi: 10.3934/dcds.2006.14.365
[15]

Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388
[16]

Yan Cui, Yanfei Wang. Velocity modeling based on Rayleigh wave dispersion curve and sparse optimization inversion. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021031
[17]

Leif Arkeryd, Raffaele Esposito, Rossana Marra, Anne Nouri. Ghost effect by curvature in planar Couette flow. Kinetic & Related Models, 2011, 4 (1) : 109-138. doi: 10.3934/krm.2011.4.109
[18]

Changfeng Gui, Huaiyu Jian, Hongjie Ju. Properties of translating solutions to mean curvature flow. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 441-453. doi: 10.3934/dcds.2010.28.441
[19]

Giulio Colombo, Luciano Mari, Marco Rigoli. Remarks on mean curvature flow solitons in warped products. Discrete & Continuous Dynamical Systems - S, 2020, 13 (7) : 1957-1991. doi: 10.3934/dcdss.2020153
[20]

Zhengchao Ji. Cylindrical estimates for mean curvature flow in hyperbolic spaces. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1199-1211. doi: 10.3934/cpaa.2021016

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences