October  2015, 8(5): 857-869. doi: 10.3934/dcdss.2015.8.857

Annihilation of two interfaces in a hybrid system

1. 

Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita ward, Sapporo, 060-0810, Japan, Japan

2. 

WPI Advanced Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan

3. 

Asahikawa Medical University, 2-1-1-1, Midorigaoka-higashi, Asahikawa 078-8510, Japan

Received  January 2014 Revised  June 2014 Published  July 2015

We consider the mixed ODE-PDE system called a hybrid system, in which the two interfaces interact with each other through a continuous medium and their equations of motion are derived in a weak interaction framework. We study the bifurcation property of the resulting hybrid system and construct an unstable standing pulse solution, which plays the role of a separator for dynamic transition from standing breather to annihilation behavior between two interfaces.
Citation: Shin-Ichiro Ei, Kei Nishi, Yasumasa Nishiura, Takashi Teramoto. Annihilation of two interfaces in a hybrid system. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 857-869. doi: 10.3934/dcdss.2015.8.857
References:
[1]

M. Argentina, P. Coullet and V. Krinsky, Head-on collisions of waves in an excitable FitzHugh-Nagumo system: a transition from wave annihilation to classical wave behavior, J. theor. Biol., 205 (2000), 47-52.

[2]

M. Argentina, P. Coullet and L. Mahadevan, Colliding waves in a model excitable medium: Preservation, annihilation, and bifurcation, Phys. Rev. Lett., 79 (1997), 2803-2806. doi: 10.1103/PhysRevLett.79.2803.

[3]

X. Chen, S.-I. Ei and M. Mimura, Self-motion of camphor discs. Model and analysis, Networks and Heterogeneous Media, 4 (2009), 1-18. doi: 10.3934/nhm.2009.4.1.

[4]

S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems, J. Dyn. Diff. Eq., 14 (2002), 85-137. doi: 10.1023/A:1012980128575.

[5]

S.-I. Ei, H. Ikeda and T. Kawana, Dynamics of front solutions in a specific reaction-diffusion system in one dimension, Jpn. J. Indust. Appl. Math., 25 (2008), 117-147. doi: 10.1007/BF03167516.

[6]

S.-I. Ei, M. Mimura and M. Nagayama, Pulse-pulse interaction in reaction-diffusion systems, Physica D, 165 (2002), 176-198. doi: 10.1016/S0167-2789(02)00379-2.

[7]

P. C. Fife, Dynamics of Internal Layers and Diffusive Interface, 1st edition, Society for Ind. and Appl. Math., Pennsylvania, 1988. doi: 10.1137/1.9781611970180.

[8]

Y. Fukao, Y. Morita and H. Ninomiya, Some entire solutions of the Allen-Cahn equation, Taiwanese J. Math., 8 (2004), 15-32.

[9]

H. Ikeda, T. Ikeda and M. Mimura, Hopf bifurcation of travelling pulses in some bistable reaction-diffusion systems, Methods and Appl. of Anal., 7 (2000), 165-193.

[10]

Y. Kuramoto, Instability and turbulence of wavefronts in reaction-diffusion systems, Prog. Theor. Phys., 63 (1980), 1885-1903. doi: 10.1143/PTP.63.1885.

[11]

Y. Morita and Y. Mimoto, Collision and collapse of layers in a 1D scalar reaction-diffusion equation, Physica D, 140 (2000), 151-170. doi: 10.1016/S0167-2789(00)00026-9.

[12]

K. Nishi, Y. Nishiura and T. Teramoto, Dynamics of two interfaces in a hybrid system with jump-type heterogeneity, Jpn. J. Ind. App. Math., 30 (2013), 351-395. doi: 10.1007/s13160-013-0100-x.

[13]

Y. Nishiura, T. Teramoto and K.-I. Ueda, Scattering and separators in dissipative systems, Phys. Rev. E, 67 (2003), 056210-1-056210-7. doi: 10.1103/PhysRevE.67.056210.

[14]

Y. Nishiura, T. Teramoto and K.-I. Ueda, Scattering of traveling spots in dissipative systems, Chaos, 15 (2005), 047509, 10pp. doi: 10.1063/1.2087127.

[15]

T. Ohta, M. Mimura and R. Kobayashi, Higher-dimensional localized patterns in excitable media, Physica D, 34 (1989), 115-144. doi: 10.1016/0167-2789(89)90230-3.

[16]

A. Scheel and J. Wright, Colliding dissipative pulses - the shooting manifold, J. Diff. Eqs., 245 (2008), 59-79. doi: 10.1016/j.jde.2008.03.019.

[17]

H. Yagisita, Backward global solutions characterizing annihilation dynamics of travelling fronts, Publ. RIMS, Kyoto Univ., 39 (2003), 117-164. doi: 10.2977/prims/1145476150.

show all references

References:
[1]

M. Argentina, P. Coullet and V. Krinsky, Head-on collisions of waves in an excitable FitzHugh-Nagumo system: a transition from wave annihilation to classical wave behavior, J. theor. Biol., 205 (2000), 47-52.

[2]

M. Argentina, P. Coullet and L. Mahadevan, Colliding waves in a model excitable medium: Preservation, annihilation, and bifurcation, Phys. Rev. Lett., 79 (1997), 2803-2806. doi: 10.1103/PhysRevLett.79.2803.

[3]

X. Chen, S.-I. Ei and M. Mimura, Self-motion of camphor discs. Model and analysis, Networks and Heterogeneous Media, 4 (2009), 1-18. doi: 10.3934/nhm.2009.4.1.

[4]

S.-I. Ei, The motion of weakly interacting pulses in reaction-diffusion systems, J. Dyn. Diff. Eq., 14 (2002), 85-137. doi: 10.1023/A:1012980128575.

[5]

S.-I. Ei, H. Ikeda and T. Kawana, Dynamics of front solutions in a specific reaction-diffusion system in one dimension, Jpn. J. Indust. Appl. Math., 25 (2008), 117-147. doi: 10.1007/BF03167516.

[6]

S.-I. Ei, M. Mimura and M. Nagayama, Pulse-pulse interaction in reaction-diffusion systems, Physica D, 165 (2002), 176-198. doi: 10.1016/S0167-2789(02)00379-2.

[7]

P. C. Fife, Dynamics of Internal Layers and Diffusive Interface, 1st edition, Society for Ind. and Appl. Math., Pennsylvania, 1988. doi: 10.1137/1.9781611970180.

[8]

Y. Fukao, Y. Morita and H. Ninomiya, Some entire solutions of the Allen-Cahn equation, Taiwanese J. Math., 8 (2004), 15-32.

[9]

H. Ikeda, T. Ikeda and M. Mimura, Hopf bifurcation of travelling pulses in some bistable reaction-diffusion systems, Methods and Appl. of Anal., 7 (2000), 165-193.

[10]

Y. Kuramoto, Instability and turbulence of wavefronts in reaction-diffusion systems, Prog. Theor. Phys., 63 (1980), 1885-1903. doi: 10.1143/PTP.63.1885.

[11]

Y. Morita and Y. Mimoto, Collision and collapse of layers in a 1D scalar reaction-diffusion equation, Physica D, 140 (2000), 151-170. doi: 10.1016/S0167-2789(00)00026-9.

[12]

K. Nishi, Y. Nishiura and T. Teramoto, Dynamics of two interfaces in a hybrid system with jump-type heterogeneity, Jpn. J. Ind. App. Math., 30 (2013), 351-395. doi: 10.1007/s13160-013-0100-x.

[13]

Y. Nishiura, T. Teramoto and K.-I. Ueda, Scattering and separators in dissipative systems, Phys. Rev. E, 67 (2003), 056210-1-056210-7. doi: 10.1103/PhysRevE.67.056210.

[14]

Y. Nishiura, T. Teramoto and K.-I. Ueda, Scattering of traveling spots in dissipative systems, Chaos, 15 (2005), 047509, 10pp. doi: 10.1063/1.2087127.

[15]

T. Ohta, M. Mimura and R. Kobayashi, Higher-dimensional localized patterns in excitable media, Physica D, 34 (1989), 115-144. doi: 10.1016/0167-2789(89)90230-3.

[16]

A. Scheel and J. Wright, Colliding dissipative pulses - the shooting manifold, J. Diff. Eqs., 245 (2008), 59-79. doi: 10.1016/j.jde.2008.03.019.

[17]

H. Yagisita, Backward global solutions characterizing annihilation dynamics of travelling fronts, Publ. RIMS, Kyoto Univ., 39 (2003), 117-164. doi: 10.2977/prims/1145476150.

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