October  2015, 8(5): 1009-1022. doi: 10.3934/dcdss.2015.8.1009

Towards modelling spiral motion of open plane curves

1. 

Department of Mathematical Sciences, School of Science and Technology, Kwansei Gakuin University, 2-1 Gakuen, Sanda 669-1337, Japan

2. 

Department of Mathematics, School of Science and Technology, Meiji University, 1-1-1 Higashi-Mita, Tama-ku, Kanagawa 214-8571

Received  January 2014 Revised  June 2014 Published  July 2015

We propose a simple evolution law for the motion of open curves with the boundary conditions towards realizing spiral growth, and derive the so-called kinematic equation. The role of the tangential velocities is studied and proved that they can be chosen arbitrarily for given boundary values. From this fact, a curvature adjusted tangential velocity for open curves is introduced. We present a numerical example which provides spiral motion starting from a line segment. This is a contrast to the case where an expanding line segment is the exact solution without the boundary conditions.
Citation: Koichi Osaki, Hirotoshi Satoh, Shigetoshi Yazaki. Towards modelling spiral motion of open plane curves. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 1009-1022. doi: 10.3934/dcdss.2015.8.1009
References:
[1]

P. K. Brazhnik, Exact solutions for the kinematic model of autowaves in two-dimensional excitable media,, Phys. D, 94 (1996), 205. doi: 10.1016/0167-2789(96)00042-5.

[2]

Y.-Y. Chen, J.-S. Guo and H. Ninomiya, Existence and uniqueness of rigidly rotating spiral waves by a wave front interaction model,, Phys. D, 241 (2012), 1758. doi: 10.1016/j.physd.2012.08.004.

[3]

V. A. Davydov, V. S. Zykov and A. S. Mikhailov, Kinematics of autowave structures in excitable media,, Sov. Phys. Usp., 34 (1991), 665.

[4]

C. L. Epstein and M. Gage, The curve shortening flow,, Wave motion: Theory, 7 (1987), 15. doi: 10.1007/978-1-4613-9583-6_2.

[5]

B. Fiedler, J.-S. Guo and J.-C. Tsai, Rotating spirals of curvature flows: A center manifold approach,, Ann. Mat. Pura Appl., 185 (2006). doi: 10.1007/s10231-004-0145-1.

[6]

J.-S. Guo, N. Ishimura and C.-C. Wu, Self-similar solutions for the kinematic model equation of spiral waves,, Phys. D, 198 (2004), 197. doi: 10.1016/j.physd.2004.08.028.

[7]

J.-S. Guo, H. Ninomiya and J.-C. Tsai, Existence and uniqueness of stabilized propagating wave segments in wave front interaction model,, Phys. D, 239 (2010), 230. doi: 10.1016/j.physd.2009.11.001.

[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. doi: 10.1007/BF03167407.

[9]

C.-P. Lo, N. S. Nedialkov and J.-M. Yuan, Classification of steady solutions of the full kinematic model,, Phys. D, 198 (2004), 258. doi: 10.1016/j.physd.2004.09.002.

[10]

A. S. Mikhailov and V. S. Zykov, Kinematical theory of spiral waves in excitable media: comparison with numerical simulations,, Phys. D, 52 (1991), 379. doi: 10.1016/0167-2789(91)90134-U.

[11]

A. S. Mikhailov, V. A. Davydov and V. S. Zykov, Complex dynamics of spiral waves and motion of curves,, Phys. D, 70 (1994), 1. doi: 10.1016/0167-2789(94)90054-X.

[12]

K. Mikula and D. Ševčovič, Evolution of plane curves driven by a nonlinear function of curvature and anisotropy,, SIAM J. Appl. Math., 61 (2001), 1473. doi: 10.1137/S0036139999359288.

[13]

K. Mikula and D. Ševčovič, Evolution of curves on a surface driven by the geodesic curvature and external force,, Appl. Anal., 85 (2006), 345. doi: 10.1080/00036810500333604.

[14]

T. Ogiwara and K.-I. Nakamura, Spiral traveling wave solutions of nonlinear diffusion equations related to a model of spiral crystal growth,, Publ. RIMS, 39 (2003), 767. doi: 10.2977/prims/1145476046.

[15]

T. Ohtsuka, Y.-H. R. Tsai and Y. Giga, A level set approach reflecting sheet structure with single auxiliary function for evolving spirals on crystal surfaces,, J. Sci. Comput., 62 (2015), 831. doi: 10.1007/s10915-014-9877-2.

[16]

P. Pauš, M. Beneš and J. Kratochvíl, Simulation of dislocation annihilation by cross-slip,, Acta Physica polonica Series A, 122 (2012), 509.

[17]

T. Sakurai, K. Osaki and T. Tsujikawa, Kinematic model of propagating arc-like segments with feedback,, Phys. D, 237 (2008), 3165. doi: 10.1016/j.physd.2008.06.001.

[18]

D. Ševčovič and S. Yazaki, Evolution of plane curves with a curvature adjusted tangential velocity,, Japan J. Indust. Appl. Math., 28 (2011), 413. doi: 10.1007/s13160-011-0046-9.

[19]

D. Ševčovič and S. Yazaki, Computational and qualitative aspects of motion of plane curves with a curvature adjusted tangential velocity,, Math. Methods in Applied Sciences, 35 (2012), 1784. doi: 10.1002/mma.2554.

[20]

D. Ševčovič and S. Yazaki, On a gradient flow of plane curves minimizing the anisoperimetric ratio,, IAENG Int. J. Appl. Math., 43 (2013), 160.

[21]

V. S. Zykov, Kinematics of wave segments moving through a weakly excitable medium,, Eur. Phys. J. Special Topics, 157 (2008), 209. doi: 10.1140/epjst/e2008-00642-x.

[22]

V. S. Zykov, Kinematics of rigidly rotating spiral waves,, Phys. D, 238 (2009), 931. doi: 10.1016/j.physd.2008.06.009.

[23]

V. S. Zykov, N. Oikawa and E. Bodenschatz, Selection of spiral waves in excitable media with a phase wave at the wave back,, Phys. Rev. Lett., 107 (2011). doi: 10.1103/PhysRevLett.107.254101.

show all references

References:
[1]

P. K. Brazhnik, Exact solutions for the kinematic model of autowaves in two-dimensional excitable media,, Phys. D, 94 (1996), 205. doi: 10.1016/0167-2789(96)00042-5.

[2]

Y.-Y. Chen, J.-S. Guo and H. Ninomiya, Existence and uniqueness of rigidly rotating spiral waves by a wave front interaction model,, Phys. D, 241 (2012), 1758. doi: 10.1016/j.physd.2012.08.004.

[3]

V. A. Davydov, V. S. Zykov and A. S. Mikhailov, Kinematics of autowave structures in excitable media,, Sov. Phys. Usp., 34 (1991), 665.

[4]

C. L. Epstein and M. Gage, The curve shortening flow,, Wave motion: Theory, 7 (1987), 15. doi: 10.1007/978-1-4613-9583-6_2.

[5]

B. Fiedler, J.-S. Guo and J.-C. Tsai, Rotating spirals of curvature flows: A center manifold approach,, Ann. Mat. Pura Appl., 185 (2006). doi: 10.1007/s10231-004-0145-1.

[6]

J.-S. Guo, N. Ishimura and C.-C. Wu, Self-similar solutions for the kinematic model equation of spiral waves,, Phys. D, 198 (2004), 197. doi: 10.1016/j.physd.2004.08.028.

[7]

J.-S. Guo, H. Ninomiya and J.-C. Tsai, Existence and uniqueness of stabilized propagating wave segments in wave front interaction model,, Phys. D, 239 (2010), 230. doi: 10.1016/j.physd.2009.11.001.

[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. doi: 10.1007/BF03167407.

[9]

C.-P. Lo, N. S. Nedialkov and J.-M. Yuan, Classification of steady solutions of the full kinematic model,, Phys. D, 198 (2004), 258. doi: 10.1016/j.physd.2004.09.002.

[10]

A. S. Mikhailov and V. S. Zykov, Kinematical theory of spiral waves in excitable media: comparison with numerical simulations,, Phys. D, 52 (1991), 379. doi: 10.1016/0167-2789(91)90134-U.

[11]

A. S. Mikhailov, V. A. Davydov and V. S. Zykov, Complex dynamics of spiral waves and motion of curves,, Phys. D, 70 (1994), 1. doi: 10.1016/0167-2789(94)90054-X.

[12]

K. Mikula and D. Ševčovič, Evolution of plane curves driven by a nonlinear function of curvature and anisotropy,, SIAM J. Appl. Math., 61 (2001), 1473. doi: 10.1137/S0036139999359288.

[13]

K. Mikula and D. Ševčovič, Evolution of curves on a surface driven by the geodesic curvature and external force,, Appl. Anal., 85 (2006), 345. doi: 10.1080/00036810500333604.

[14]

T. Ogiwara and K.-I. Nakamura, Spiral traveling wave solutions of nonlinear diffusion equations related to a model of spiral crystal growth,, Publ. RIMS, 39 (2003), 767. doi: 10.2977/prims/1145476046.

[15]

T. Ohtsuka, Y.-H. R. Tsai and Y. Giga, A level set approach reflecting sheet structure with single auxiliary function for evolving spirals on crystal surfaces,, J. Sci. Comput., 62 (2015), 831. doi: 10.1007/s10915-014-9877-2.

[16]

P. Pauš, M. Beneš and J. Kratochvíl, Simulation of dislocation annihilation by cross-slip,, Acta Physica polonica Series A, 122 (2012), 509.

[17]

T. Sakurai, K. Osaki and T. Tsujikawa, Kinematic model of propagating arc-like segments with feedback,, Phys. D, 237 (2008), 3165. doi: 10.1016/j.physd.2008.06.001.

[18]

D. Ševčovič and S. Yazaki, Evolution of plane curves with a curvature adjusted tangential velocity,, Japan J. Indust. Appl. Math., 28 (2011), 413. doi: 10.1007/s13160-011-0046-9.

[19]

D. Ševčovič and S. Yazaki, Computational and qualitative aspects of motion of plane curves with a curvature adjusted tangential velocity,, Math. Methods in Applied Sciences, 35 (2012), 1784. doi: 10.1002/mma.2554.

[20]

D. Ševčovič and S. Yazaki, On a gradient flow of plane curves minimizing the anisoperimetric ratio,, IAENG Int. J. Appl. Math., 43 (2013), 160.

[21]

V. S. Zykov, Kinematics of wave segments moving through a weakly excitable medium,, Eur. Phys. J. Special Topics, 157 (2008), 209. doi: 10.1140/epjst/e2008-00642-x.

[22]

V. S. Zykov, Kinematics of rigidly rotating spiral waves,, Phys. D, 238 (2009), 931. doi: 10.1016/j.physd.2008.06.009.

[23]

V. S. Zykov, N. Oikawa and E. Bodenschatz, Selection of spiral waves in excitable media with a phase wave at the wave back,, Phys. Rev. Lett., 107 (2011). doi: 10.1103/PhysRevLett.107.254101.

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