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 & 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.

[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. 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. 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. 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. 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. doi: 10.1016/S0167-2789(02)00379-2.

[7]

P. C. Fife, Dynamics of Internal Layers and Diffusive Interface,, 1st edition, (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.

[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.

[10]

Y. Kuramoto, Instability and turbulence of wavefronts in reaction-diffusion systems,, Prog. Theor. Phys., 63 (1980), 1885. 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. 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. 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. 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). 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. 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. doi: 10.1016/j.jde.2008.03.019.

[17]

H. Yagisita, Backward global solutions characterizing annihilation dynamics of travelling fronts,, Publ. RIMS, 39 (2003), 117. 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.

[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. 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. 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. 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. 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. doi: 10.1016/S0167-2789(02)00379-2.

[7]

P. C. Fife, Dynamics of Internal Layers and Diffusive Interface,, 1st edition, (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.

[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.

[10]

Y. Kuramoto, Instability and turbulence of wavefronts in reaction-diffusion systems,, Prog. Theor. Phys., 63 (1980), 1885. 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. 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. 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. 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). 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. 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. doi: 10.1016/j.jde.2008.03.019.

[17]

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

[1]

Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631

[2]

Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245

[3]

Rebecca McKay, Theodore Kolokolnikov, Paul Muir. Interface oscillations in reaction-diffusion systems above the Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2523-2543. doi: 10.3934/dcdsb.2012.17.2523

[4]

M. Grasselli, V. Pata. A reaction-diffusion equation with memory. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1079-1088. doi: 10.3934/dcds.2006.15.1079

[5]

Lu Yang, Meihua Yang. Long-time behavior of stochastic reaction-diffusion equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2627-2650. doi: 10.3934/dcdsb.2017102

[6]

Yasumasa Nishiura, Takashi Teramoto, Xiaohui Yuan. Heterogeneity-induced spot dynamics for a three-component reaction-diffusion system. Communications on Pure & Applied Analysis, 2012, 11 (1) : 307-338. doi: 10.3934/cpaa.2012.11.307

[7]

Toshiyuki Ogawa, Takashi Okuda. Oscillatory dynamics in a reaction-diffusion system in the presence of 0:1:2 resonance. Networks & Heterogeneous Media, 2012, 7 (4) : 893-926. doi: 10.3934/nhm.2012.7.893

[8]

Maria do Carmo Pacheco de Toledo, Sergio Muniz Oliva. A discretization scheme for an one-dimensional reaction-diffusion equation with delay and its dynamics. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1041-1060. doi: 10.3934/dcds.2009.23.1041

[9]

Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382

[10]

Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure & Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229

[11]

Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493

[12]

Yuncheng You. Asymptotic dynamics of reversible cubic autocatalytic reaction-diffusion systems. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1415-1445. doi: 10.3934/cpaa.2011.10.1415

[13]

Wenzhang Huang, Maoan Han, Kaiyu Liu. Dynamics of an SIS reaction-diffusion epidemic model for disease transmission. Mathematical Biosciences & Engineering, 2010, 7 (1) : 51-66. doi: 10.3934/mbe.2010.7.51

[14]

Liang Zhang, Zhi-Cheng Wang. Threshold dynamics of a reaction-diffusion epidemic model with stage structure. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3797-3820. doi: 10.3934/dcdsb.2017191

[15]

Shin-Ichiro Ei, Toshio Ishimoto. Effect of boundary conditions on the dynamics of a pulse solution for reaction-diffusion systems. Networks & Heterogeneous Media, 2013, 8 (1) : 191-209. doi: 10.3934/nhm.2013.8.191

[16]

Hongyan Zhang, Siyu Liu, Yue Zhang. Dynamics and spatiotemporal pattern formations of a homogeneous reaction-diffusion Thomas model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1149-1164. doi: 10.3934/dcdss.2017062

[17]

Xiaoyan Zhang, Yuxiang Zhang. Spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-16. doi: 10.3934/dcdsb.2018124

[18]

W. E. Fitzgibbon, M. Langlais, J.J. Morgan. A reaction-diffusion system modeling direct and indirect transmission of diseases. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 893-910. doi: 10.3934/dcdsb.2004.4.893

[19]

José-Francisco Rodrigues, Lisa Santos. On a constrained reaction-diffusion system related to multiphase problems. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 299-319. doi: 10.3934/dcds.2009.25.299

[20]

Haomin Huang, Mingxin Wang. The reaction-diffusion system for an SIR epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2039-2050. doi: 10.3934/dcdsb.2015.20.2039

2016 Impact Factor: 0.781

Metrics

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

[Back to Top]