2010 Fiscal Year Final Research Report
Mathematical Theory of Nonlinear-Non-equilibrium Reaction-Diffusion Systems
| Project/Area Number |
18104002
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| Research Category |
Grant-in-Aid for Scientific Research (S)
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| Allocation Type | Single-year Grants |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Meiji University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
NISHIURA Yasumasa 北海道大学, 電子科学研究所, 教授 (00131277)
YANAGIDA Eiji 東北大学, 大学院・理学研究科, 教授 (80174548)
MATANO Hiroshi 東京大学, 大学院・数理科学研究科, 教授 (40126165)
KOBAYASHI Ryo 広島大学, 大学院・理学研究科, 教授 (60153657)
EI Shin-ichiro 九州大学, 数理学研究院, 教授 (30201362)
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| Project Period (FY) |
2006 – 2010
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| Keywords | 反応拡散方程式 / 非線形非平衡現象 / 自己組織化 / 進行波 / フロント波/スポット波の相互作用 / 解の爆発 / 特異極限解析 / 無限次元力学系 / 大域的分岐理論 |
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| Research Abstract |
Reaction diffusion equations already appeared as mathematics models which described population genetics, ecology and so on from the early 20^<th> century, and the qualitative study was performed in the world of mathematics. In the late 20^<th> century, there was a great breakthrough in sciences, that is, nonlinear non-equilibrium science and reaction diffusion equations appeared in natural ecienc such as physics, chemistry, biology and other fields, as mathematics models describing various nonlinear non-equilibrium phenomena. Thus, the study of reaction diffusion system has been pushed forward not only in mathematics but also in widely natural sciences. The result in this study was able to establish the analytical technique of spatio-temporal patterns arising in reaction diffusion equations from viewpoints of mathematics and applied mathematics for mathematical elucidation of the nonlinear-non-equilibrium phenomenon. As examples, there are (1) the construction of "the invariant manifold theory of infinite dimension dynamical systems" to handle pattern dynamics appearing as dissipative structure and self-organization, (2) the construction of "analytical theory of transient pattern dynamics" to understand the transition process from a simple pattern to a complex one, which is a typical nonlinear-non-equilibrium phenomenon, and (3) the construction of "the singular limit theory" to understand complex dynamic patterns and stationary forms in nonlinear non-equilibrium systems. These results enable us to mathematically understand spatio-temporal patterns in nonlinear non-equilibrium systems.
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