激励介质的非线性波型动力学(英文)

doi:10.7523/j.issn.2095-6134.2004.2.020

摘要/Abstract

摘要：

以刻划著名的Belousov Zhabotinsky化学反应的俄勒冈振子为数学模型,研究解决了激励介质中一些悬而未决的理论问题 (如波的存在性和稳定性等),进一步完善了激励介质的非线性波型动力学的理论体系.通过Painlev啨分析,B¨acklund变换和奇异摄动方法,分析地给出了一些常见的波型解 (如行波,螺旋波,靶型波,V 型波,涡卷波等).在波前的邻域内,通过引进新的运动坐标系,获得了波在直角坐标下的运动方程.特别是定量地给出了刻划小幅波的组织中心沿轴向和径向运动的规律,并由此可判定波的组织中心何时沿径向呈现膨胀或收缩,何时沿轴向有正向或反向漂移.这一结果很好地与实验和数值模拟结果相吻合.此外,研究了耦合的俄勒冈振子,解决了Tyson于 1 979年提出的一个猜想 (即对耦合的俄勒冈振子,稳定的回声波可以和稳定的正定态共存),提出了一套解决类似问题的一般方法.

关键词: 激励介质, 俄勒冈振子, BZ化学反应, 波型解, 组织中心

Abstract:

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.

Key words: excitable media, Oregonator, Belousov-Zhabotinsky chemical reaction, wave pattern solution, organizing filament

中图分类号:

周天寿, 张锁春. 激励介质的非线性波型动力学(英文)[J]. 中国科学院大学学报, 2004, 21(2): 276-281.

ZHOU Tian-Shou, ZHANG Suo-Chun. Dynamics of Nonlinear Wave Patternin Excitable Media[J]. , 2004, 21(2): 276-281.

参考文献

[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