# American Institute of Mathematical Sciences

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 and 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), 205-220. 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), 1758-1766. 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), 665-684. [4] C. L. Epstein and M. Gage, The curve shortening flow, Wave motion: Theory, modelling, and computation (Berkeley, Calif., 1986), Math. Sci. Res. Inst. Publ., Springer, New York, 7 (1987), 15-59. 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), suppl., S259-S291. 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), 197-211. 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), 230-239. 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), 317-330. 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), 258-280. 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), 379-397. 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), 1-39. 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), 1473-1501. 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), 345-362. 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, Kyoto Univ., 39 (2003), 767-783. 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), 831-874. 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), 509-511. [17] T. Sakurai, K. Osaki and T. Tsujikawa, Kinematic model of propagating arc-like segments with feedback, Phys. D,237 (2008), 3165-3171. 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), 413-442. 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), 1784-1798. 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), 160-171. [21] V. S. Zykov, Kinematics of wave segments moving through a weakly excitable medium, Eur. Phys. J. Special Topics, 157 (2008), 209-221. doi: 10.1140/epjst/e2008-00642-x. [22] V. S. Zykov, Kinematics of rigidly rotating spiral waves, Phys. D, 238 (2009), 931-940. 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), 254101. doi: 10.1103/PhysRevLett.107.254101.

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##### References:
 [1] P. K. Brazhnik, Exact solutions for the kinematic model of autowaves in two-dimensional excitable media, Phys. D, 94 (1996), 205-220. 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), 1758-1766. 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), 665-684. [4] C. L. Epstein and M. Gage, The curve shortening flow, Wave motion: Theory, modelling, and computation (Berkeley, Calif., 1986), Math. Sci. Res. Inst. Publ., Springer, New York, 7 (1987), 15-59. 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), suppl., S259-S291. 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), 197-211. 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), 230-239. 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), 317-330. 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), 258-280. 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), 379-397. 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), 1-39. 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), 1473-1501. 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), 345-362. 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, Kyoto Univ., 39 (2003), 767-783. 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), 831-874. 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), 509-511. [17] T. Sakurai, K. Osaki and T. Tsujikawa, Kinematic model of propagating arc-like segments with feedback, Phys. D,237 (2008), 3165-3171. 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), 413-442. 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), 1784-1798. 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), 160-171. [21] V. S. Zykov, Kinematics of wave segments moving through a weakly excitable medium, Eur. Phys. J. Special Topics, 157 (2008), 209-221. doi: 10.1140/epjst/e2008-00642-x. [22] V. S. Zykov, Kinematics of rigidly rotating spiral waves, Phys. D, 238 (2009), 931-940. 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), 254101. doi: 10.1103/PhysRevLett.107.254101.
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