Dissertation > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations

Research on Dynamics of Nonlinear Wave Pattern in Excitable Media

Author ZhouTianShou
Tutor ZhangSuoChun
School Mathematics and Systems Science Institute
Course Applied Mathematics
Keywords Oregonator Model Excitable Medium Organizing Filament Anti-Phase Wave Tyson’s Conjecture
CLC O175
Type PhD thesis
Year 2001
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In this thesis we work for theoretical study on dynamics of nonlinear wave pattern in excitable media by taking the Oregonator model portraying Belousov-Zhabotinsky chemical reaction as an example. Although there have thus far existed many experimental reports and numerical simulation results for it, the theoretical system about the aspect is fairly incomplete and there exist some outstanding theoretical questions. We comparatively comprehensively give various wave pattern solutions, and obtain laws of motion of waves (including motion of wave fronts and organizing filaments); on the other hand, we study such some basic problems as existence and stability of the waves. In a word, all our results further perfect the theoretical system of the dynamics of the nonlinear wave patterns in the excitable media. The detailed contents are as follows:In chapter two, We use Painlevé analysis technic to first obtain explicit expressions of traveling waves or planar waves and dispersion relation as well as expressions of series of the corresponding wave pattern solutions within thin boundary layer. The technic is easily extended to other analogous reaction-diffusion equations. In addition we establish a new moving coordinate system in the neighborhood of the wave front and obtain a revised eikonal equation describing the curvature effect of waves in the usual orthogonal coordinate system. The equation of the form overcomes the shortcomings to obtain difficultly the wave pattern solutions directly from its original form and has obviously such advantages as easily being solved and containing some usual wave pattern solutions.In chapter three, by applying Ba|¨cklund transformations we give the approximate explicit solutions of traveling waves in the Oregonator model. For the small amplitude waves.

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