Misc - Ei Shin-Ichiro
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Heterogeneity-induced effects for pulse dynamics in FitzHugh–Nagumo-type systems Reviewed
Chao Nien Chen, Shin Ichiro Ei, Shin Ichiro Ei, Shyuh yaur Tzeng
Physica D: Nonlinear Phenomena 2018.01
Language：English
© 2018 Elsevier B.V. Particle like structures have been observed in many fields of science. In a homogeneous medium, a stable, standing pulse is a localized wave that may arise when nonlinear and dissipative effects are in balance. In this paper, we investigate certain phenomena associated with the dynamics of pulse solutions for a FitzHugh–Nagumo reaction–diffusion model. When two pulses are located far from one another initially, their weak interaction drives the subsequent slow dynamics. Our comprehension of the standing pulse profiles allows us to quantitatively characterize their interplay; when the diffusivity of the activator is small compared to that of the inhibitor, the two pulses repel. In addition, using a center-manifold reduction to study the presence of heterogeneities in the environment, we demonstrate that the pulses will move so as to maximize the strength of activation or minimize that of inhibition. The pulse motion will also be influenced by the reaction mechanism.
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Dynamics of localized solutions for reaction-diffusion systems on two dimensional domain : Spot dynamics on curved surface (Nonlinear Partial Differential Equations, Dynamical Systems and Their Applications)
Ei Shin-Ichiro, Yagisita Hiroki
RIMS Kokyuroku 1881 66 - 70 2014.04
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Dynamics of Spot Solutions in Two Dimensional Spaces(Invited Lectures in the JSIAM Annual Meeting 2013)
Ei Shin-Ichiro
応用数理 24 ( 1 ) 34 - 36 2014.03
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Mathematical Analysis for Pattern Formation Problems,A Mathematical Approach to Research Problems of Science and Technology
R. Nishii, S.-I. Ei, M. Koiso, H. Ochiai, K. Okada, S. Saito, T. Shirai, Editors
Mathematics for Industry 5, Springer 2014 133 - 139 2014
解説・総説
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A NEW TREATMENT FOR PERIODIC SOLUTIONS AND COUPLED OSCILLATORS
Shin-Ichiro Ei, Kunishige Ohgane
KYUSHU JOURNAL OF MATHEMATICS 65 ( 2 ) 197 - 217 2011.09
Language：English Publisher：KYUSHU UNIV, FAC MATHEMATICS
We develop a systematic method for deriving the phase dynamics of perturbed periodic solutions. The method is to regard periodic solutions as slowly modulated traveling solutions on the circle. There, problems are reduced to the perturbed problems from stationary solutions on the circle. This makes the treatment of periodic solutions far easier and systematic. We also give the rigorous proofs for this method.
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Interactive dynamics of two interfaces in a reaction diffusion system (Nonlinear Evolution Equations and Mathematical Modeling)
Ei Shin-Ichiro, Tsujikawa Tohru
RIMS Kokyuroku 1588 118 - 123 2008.04
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25aVII-1 Derivation of phase dynamics of perturbed periodic solutions and the applications
Ei Shin-Ichiro
Meeting abstracts of the Physical Society of Japan 63 ( 1 ) 302 - 302 2008.02
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Dynamics of front solutions in a specific reaction-diffusion system in one dimension
Shin -Ichiro Ei, Hideo Ikeda, Takeyuki Kawana
Japan Journal of Industrial and Applied Mathematics 25 ( 1 ) 117 - 147 2008.02
Language：English Publisher：Springer Science and Business Media LLC
In this paper, two component reaction-diffusion systems with a specific bistable nonlinearity are concerned. The systems have the bifurcation structure of pitch-fork type of traveling front solutions with opposite velocities, which is rigorously proved and the ordinary differential equations describing the dynamics of such traveling front solutions are also derived explicitly. It enables us to know rigorously precise information on the dynamics of traveling front solutions. As an application of this result, the imperfection structure under small perturbations and the dynamics of traveling front solutions on heterogeneous media are discussed.
DOI： 10.1007/bf03167516
Other Link： http://link.springer.com/article/10.1007/BF03167516/fulltext.html
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Dynamics of front solutions in a specific reaction-diffusion system in one dimension Reviewed
Shin-Ichiro Ei, Hideo Ikeda, Takeyuki Kawana
JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS 25 ( 1 ) 117 - 147 2008.02
Language：English Publisher：KINOKUNIYA CO LTD
In this paper, two component reaction-diffusion systems with a specific bistable nonlinearity are concerned. The systems have the bifurcation structure of pitch-fork type of traveling front solutions with opposite velocities, which is rigorously proved and the ordinary differential equations describing the dynamics of such traveling front solutions are also derived explicitly. It enables us to know rigorously precise information on the dynamics of traveling front solutions. As an application of this result, the imperfection structure under small perturbations and the dynamics of traveling front solutions on heterogeneous media are discussed.
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Interacting spots in reaction diffusion systems Reviewed
SI Ei, M Mimura, M Nagayama
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS 14 ( 1 ) 31 - 62 2006.01
Authorship：Lead author Language：English Publisher：AMER INST MATHEMATICAL SCIENCES
This paper is concerned with the dynamics of travelling spot solutions in two dimensions. Travelling spot solutions are constructed under the bifurcation structure with Jordan block type degeneracy. It is shown that if the velocity is very slow, such travelling spots possess reflection property. In order to do it, we derive the reduced ordinary differential equations describing the dynamics of interacting travelling spots in RD systems by using center manifold theory. This reduction enables us to prove that two very slowly travelling spots reflect before collision as if they were elastic particles.
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Dynamics of Turing Patterns for Reaction-Diffusion Systems in a Cylindrical Domain on 2D, Proceedings of Hyperbolic problems,Theory
栄 伸一郎
Numerics and Applications 2004(eds. Asakura, Aiso, Kawashima,Matsumura, Nishibata, Nishihara) 121 - 128 2006
Yokohama Publishers<br />
解説・総説 -
A variational approach to singular perturbation problems in reaction-diffusion systems Reviewed
SI Ei, M Kuwamura, Y Morita
PHYSICA D-NONLINEAR PHENOMENA 207 ( 3-4 ) 171 - 219 2005.08
Language：English Publisher：ELSEVIER SCIENCE BV
In this paper singular perturbation problems in reaction-diffusion systems are studied from a viewpoint of variational principle. The goal of the study is to provide an unified and transparent framework to understand existence, stability and dynamics of solutions with transition layers in contrast to previous works in many literatures on singular perturbation theory. (c) 2005 Elsevier B.V. All rights reserved.
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Dynamics of Turing patterns in cylindrical domains on 2D (Mathematical Analysis in Fluid and Gas Dynamics)
Ei Shin-Ichiro
RIMS Kokyuroku 1425 122 - 129 2005.04
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Dynamics of Turing patterns in cylindrical domains in 2D,流体と気体の数学解析
栄 伸一郎
数理解析研究所講究録 1425 122 - 129 2005
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2次元帯状領域におけるパターンの運動 (反応拡散系におけるパターン形成と漸近的幾何構造の研究)
栄 伸一郎
数理解析研究所講究録 1356 108 - 115 2004.02
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Dynamics of metastable localized patterns and its application to the interaction of spike solutions for the Gierer-Meinhardt systems in two spatial dimensions Reviewed
SI Ei, JC Wei
JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS 19 ( 2 ) 181 - 226 2002.06
Language：English Publisher：SPRINGER JAPAN KK
In this paper, the Gierer-Meinhardt model systems with finite diffusion constants in the whole space R-2 is considered. We give a regorous proof on the existence and the stability of a single spike solution, and by using such informations, the repulsive dynamics of the interacting multi single-spike solutions is also shown when distances between spike solutions are sufficiently large. This clarifies some part of the mechanism of the evolutional process of localized patterns appearing in the Gierer-Meinhardt model equations.
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Dynamics of pulse-like localized solutions in reaction-diffusion systems (International Conference on Reaction-Diffusion Systems : Theory and Applications)
Ei Shin-Ichiro, Mimura M
RIMS Kokyuroku 1249 9 - 17 2002.02
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Shin-Ichiro Ei, Juncheng Wei
Japan Journal of Industrial and Applied Mathematics 19 ( 2 ) 181 - 226 2002
Language：English Publisher：Kinokuniya Co. Ltd
In this paper, the Gierer-Meinhardt model systems with finite diffusion constants in the whole space R2 is considered. We give a regorous proof on the existence and the stability of a single spike solution, and by using such informations, the repulsive dynamics of the interacting multi single-spike solutions is also shown when distances between spike solutions are sufficiently large. This clarifies some part of the mechanism of the evolutional process of localized patterns appearing in the Gierer-Meinhardt model equations.
DOI： 10.1007/BF03167453
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2(n)-splitting or edge-splitting? A manner of splitting in dissipative systems Reviewed
S Ei, Y Nishiura, K Ueda
JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS 18 ( 2 ) 181 - 205 2001.06
Language：English Publisher：KINOKUNIYA CO LTD
Since early 90's, much attention has been paid to dynamic dissipative patterns in laboratories, especially, self-replicating pattern (SRP) is one of the most exotic phenomena. Employing model system such as the Gray-Scott model, it is confirmed also by numerics that SRP can be obtained via destabilization of standing or traveling spots. SRP is a typical example of transient dynamics, and hence it is not a priori clear that what kind of mathematical framework is appropriate to describe the dynamics. A framework in this direction is proposed by Nishiurar-Ueyama [16], i.e., hierarchy structure of saddle-node points, which gives a basis for rigorous analysis. One of the interesting observation is that when there occurs self-replication, then only spots (or pulses) located at the boundary (or edge) are able to split. Internal ones do not duplicate at all. For ID-case, this means that the number of newly born pulses increases like 2k after k-th splitting, not 2(n)-splitting where all pulses split simultaneously. The main objective in this article is two-fold: One is to construct a local invariant manifold near the onset of self-replication, and derive the nonlinear ODE on it. The other is to study the manner of splitting by analysing the resulting ODE, and answer the question "2(n)-splitting or edge-splitting?" starting from a single pulse. It turns out that only the edge-splitting occurs, which seems a natural consequence from a physical point of view, because the pulses at edge are easier to access fresh chemical resources than internal ones.