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Staff Introduction

EI, Shin-Ichiro / Guest Professor

The understanding of various patterns such as snow crystal, combustion, spot patterns of groups of plankton, and other kinds of chemical patterns appearing in nature are one of the most attractive objects of study in natural science. My interest is to theoretically study the structure and mechanism of such phenomena. To do it, I use description through model equations, which is one of the most theoretical methods, well known since Newton. The model equations which I have studied so far lie in the framework of partial differential equations which describe evolutional processes of certain materials with two mechanisms: (1) the diffusion process in space, and (2) the production and/or extinction of materials. Such model equations are generally called "reaction-diffusion systems" and their use has been well recognized in physics, chemistry, biology and other fields from a mathematical modelling point of view. In order to analyze these models, a powerful tool called the integral manifold theory has been recently developed to study the evolutional process of the equations. Mathematically speaking, this theory proposes to construct a certain set with properties if a solution starts with the initial data in the set, the solution also remains in the set for an any time. Such a set is called invariant manifold. Recently, under appropriate conditions, I have derived the simpler ordinary differential equations governing solutions on an invariant manifold, so that much information on solutions of original reaction diffusion systems can be obtained by solving the reduced ordinary differential equations. This is called multi-time scale method in applied mathematics. Thus coupling of the integral manifold theory and multi-time scale methods is very useful in studying reaction diffusion systems and, in practice, a lot of information on mechanism of various phenomena has been obtained.

Keywords Nonlinear Analysis
Faculty , Department Institute of Mathematics for Industry , Visitors Section