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Shin-Ichiro Ei

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Non-Linear Analysis and its Applications

Shin-Ichiro Ei
Degree: Doctor of Science (Hiroshima University)
Research Interests: Non-Linear Analysis
Unit: Space & Flow

Report

Complicated temporal and spatial patterns are frequently seen in nature. These patterns often appear in so called dissipative system, the processes of continuous inflow and outflow of energy, and can be seen in life phenomena, including the maintenance of morphology by breathing and consuming food or the characteristic pattern maintained during a chemical reaction. The simplest dissipative patterns can be described using a model equation called the reaction-diffusion equation. In 1952, Turing theoretically suggested a mechanism of morphogenesis in living organisms using a simple model. This mechanism is now known as diffusion induced instability or Turing instability. Prigogine then expanded this idea to include various other dissipative systems, and established a theory of dissipative structures. For his work, Prigogine was awarded the Nobel Prize in chemistry in 1977.

On the other hand, although it was considered almost impossible to apply these theories to the morphology of really living organisms, Kondo et. al. of Japan proved this notion wrong. Actually when in 1995, they demonstrated that not only the pattern on the body surface of a certain tropical fish could be described using a reaction-diffusion type model, but also that the changes in the pattern as the fish grows are faithfully recreated. This discovery has elucidated the current understanding that the mechanism of diffusion induced instability is fundamentally responsible for the process of morphogenesis in living organisms. Preceding Kondo et.al in 1991, a group in Bordeaux, France led by de Kepper demonstrated that in a chemical reaction system using a gel called Chlorite-Iodite-Malonic Acid(CIMA), a stripe pattern and a polka dot pattern appear by the diffusion induced instability mechanism.

Thus, it can be concluded that diffusion induced instability is a universal mechanism responsible for spontaneous pattern formation in living organisms and chemical reaction systems. Spontaneous pattern formation is also called self-organization, and has received much attention for applied nanotechnologies. In fact, certain carbon nanotubes are considered to be formed via a self-organized mechanism, and several reaction-diffusion models have also been proposed. As seen above, pattern formation theories such as the diffusion induced instability are becoming more important not only in natural science, but also in industry.

The focus of my group’s research is mathematical analysis of reaction-diffusion type equations among the above topics. My research interest is the theoretical pursuit of the behavior of various solutions, such as pattern formation. Typical spatial patterns expressed by reaction-diffusion equations include pulse-like localized patterns and interfaces expressed as contour lines (surfaces) of solutions. Examples of the former include the transmission of neural impulses along a nerve axon and spiky localized patterns in a model of morphogenesis, while examples of the latter include crystal shapes in solidification phenomena and the shapes of flames while burning.

Figure Formation and development of spiky localized patterns in a morphogenesis model

My research has helped establish fundamental theories, which theoretically consider interactions when multiple pulse-like localized patterns exist. Other contributions include the proposition of a method to consider the stability of a standing interface. My group’s efforts have provided effective analytical methods for researching the temporal development of spatial patterns, and our applications include repulsive interactions of spiky localized patterns and stability analysis of membrane interfaces with an applied pressure.

Similar to patterns in real-world phenomena, patterns of solutions to model equations are often extremely complex. As a result, theoretical analysis alone have their limitations, which make computer simulations a very powerful research tool in understanding the behaviors of solutions. Therefore, students in my group must be able to conduct a numerical simulation study on their own as well as be able to perform mathematical analysis. Graduates from our group are capable of understanding phenomena and models mathematically as well as conduct computer simulations, and they are currently working in a variety of fields, including data communications, software development, education, and research.

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