2018 Fiscal Year Final Research Report
Relation between pattern formations and complex singularities of solutions of nonlinear partial differential equations
| Project/Area Number |
16K13778
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| Research Category |
Grant-in-Aid for Challenging Exploratory Research
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| Allocation Type | Multi-year Fund |
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| Research Field |
Foundations of mathematics/Applied mathematics
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| Research Institution | Meiji University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
桂田 祐史 明治大学, 総合数理学部, 専任准教授 (80224484)
池田 幸太 明治大学, 総合数理学部, 専任准教授 (50553369)
小野寺 有紹 東京工業大学, 理学院, 准教授 (70614999)
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| Project Period (FY) |
2016-04-01 – 2019-03-31
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| Keywords | 反応拡散系 / 複素特異点 / パターン形成 / 自由境界問題 |
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| Outline of Final Research Achievements |
We showed the analytic extension of the solution of a reaction-diffusion equation into complex domain. However, in general, the complex singularities appear. To study the motion of the singularities, we consider the heat equation and the Allen-Cahn-Nagumo equation. For the Allen-Cahn-Nagumo equation, we study the motion of the complex singularities of the exact solutions. We tried to extend the properties derived from this observation into the case for general nonlinearity.
Moreover, we study the bifurcation from infinity and we get the result which generalizes the results of Stuart and Rabinowitz into the multi-component reaction-diffusion systems.
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| Free Research Field |
非線形偏微分方程式
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| Academic Significance and Societal Importance of the Research Achievements |
非線形偏微分方程式は，天気予報などの身近な問題だけでなく，ナノテクノロジーから宇宙の解明まで幅広い分野で利用されている．しかし，非線形偏微分方程式の解を表現する解の公式はないため，解の形状を表現する手法の開発が求められている．本研究課題では，複素特異点や無限遠からのパターン形成について考察し，一部は論文として投稿するに至った．
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