Journal of University of Chinese Academy of Sciences >
Dynamics of Nonlinear Wave Patternin Excitable Media
Received date: 2003-09-08
Online published: 2004-03-19
Based on the Oregonator model portraying the famous Belousov Zhabotinsky chemical reaction,the dynamics of nonlinear wave pattern in excitable media is further consummated. Some basic problems such as existence and stability of waves are studied. By Painlevéanalysis,B?cklund transformationand pertubation method,some usual wave pattern solutions (for example, travelling wave,spiral wave,target wave,Vtype wave,scroll wave,etc) are analytically presented.By establishing new moving coordinatesystems in the neighborhood of the wave fornt,equations of motion of waves which describe the curvatureeffect of waves,are derived in the orthogonal coordinate system. In particular,lawof motion of the organizing filament along the radial and axial directions,respectively,is quantitatively obtained. It demonstrateswhether the filament expands or shrinks along the radial direction,and positively or inversely shifts alongthe axial direction. The result is in good accord with the existing experimental data. In addition,the coupled Oregonator is studied. Tyson’s conjecture that the stable homogeneous positive steady state may coexist with the stable echo wave in the coupled model is basically solved. The corresponding proof procedurepractically gives a general method in handling other similar problems.
ZHOU Tian-Shou , ZHANG Suo-Chun . Dynamics of Nonlinear Wave Patternin Excitable Media[J]. Journal of University of Chinese Academy of Sciences, 2004 , 21(2) : 276 -281 . DOI: 10.7523/j.issn.2095-6134.2004.2.020
[1] Zaikin A N,Zhabotinsky A M.Concentration wave propagation in two-dimensional liquid-phase self-oscillation system.Nature,1970,225 :535-537
[2] Winfree A T.Wavelike activity in biological and chemical media in lecture notes in biomath ematics.P Van Der Driesshe,ed.Berlin :Springer-Ver-lag,1974
[3] Winfree A T,Jahnke W.Three-dimensional scroll ring dynamics in the Belousov-Zhabotinsky reaction and in the two-variable Oregonator model.J.Phys.Chem.,1989,93 :2823-2832
[4] Winfree A T.Stable particle-like solutions to the nonlinear wave equations of three-dimensional excitable media.SIAM Review,1990,32 (1) :1-53
[5] Karma A.Scaling regim of spiral wave propagation in single-diffusion excitable.Phys.Rev.Lett.,1992,68 (3) :397-400
[6] Keener J P,Tyson J J.The motion of untwisted untorted scroll waves in Belousov-Zhabotinsky reagent.Science,1988,239 :1284-1286
[7] Keener J P.A geometrical theory for spiral waves inn excitable media.SIAM J.Appl.Math.,1986,46 (6) :1039-1056
[8] Mikhailov A S,Krinsky V I.Rotating spiral waves in excitable media :the analytical results.Physica D,1983,9 :346-371
[9] GomatamJ,Grindrod P.Three-dimensional waves in excitable reaction-diffusion systems.J.Math.Biol.,1987,25 :611-622
[10] Zhou T S,He XL.Existence and linear stability of planar waves in the Oregonator model.Chinese Phys.,2002,11 (12) :1234-1239
[11] Ding D F.A plausible mechanism for the motion of untwisted scroll rings in excitable media.Physica D,1988,32 :471-487
[12] Zhou T S.Law of motion of waves in excitable media.Int.J.Theor.Phys.,Group Theor.and Nonlinear Optics,2003,10 (1) :56-61
[13] Zhou T S,Zhang S C.Equation of motion of wave in excitable media.Acta Math.Appl.Sinica,2000,23 (4) :596-602 (in Chinese)
[14] Zhou T S,Zhang S C,Zheng Z H.Analytical representation of travelling wave solutions and dispersive relation in Tyson’s model.Comm.Theor.Phys.,2000,33 (1) :147-150
[15] Zhou T S,Zhang S C.Explicit representation of planar wave speed and dispersive relation in general excitable media.Chin.Phys.,2000,9 (3) :166-170
[16] Keener J P.The dynamics of three-dimensional scroll waves in excitable media.Physica D,1988,31 :269-276
[17] Keener J P,Tyson J J.The dynamics of helical scroll waves in excitable media.Physica D,1991,53 :151-161
[18] Biktashev V N,Holden A V,Zhang H.Tension of organizing filaments of scroll waves.Phil.Trans.R.Soc.London A.,1994,347 :611-630
[19] Zhou T S.Law of motion of scroll amplitude scroll wave filament.Int.J.Theor.Phys.,Group Theor.and Nonlinear Optics,2003,9 (4) :287-303
[20] Zhou T S,Zhang S C.Organzing filament of small amplitude scroll ring.Comm.Theor.Phys.,2001,35 (6) :657-660
[21] Zhou T S,Zhang S C,Zheng Z H.The motion law of the scroll wave filament.Comm.Theor.Phys.,2000,34 (1) :185-188
[22] Tyson J J.Bistability and echo waves in models of the Belousov-Zhabotinsky reaction.Ann.N.Y.Acad.Sci.,1979,316 :279-295
[23] Zhou T S,Zhang S C.Dynamical behavior in linearly coupled Oregonators.Physica D,2001,151 :199-216
[24] Zhou T S,Zhang S C.Echo waves and coexisting phenomenon in coupled Brusselators.Chaos,Solitons and Fractals,2002,13 :621-632
[25] Zheng Z H,Zhou T S,Zhang S C.Existence of echo waves in the coupled Oregonator.Chaos,Solitons and Fractals,2002,14 :425-432
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