Shin-Ichiro Ei

The Motion of Weakly Interacting Pulses in Reaction-Diffusion Systems

The interaction of stable pulse solutions on R1 is considered when distances between pulses are sufficiently large. We construct an attractive local invariant manifold giving the dynamics of interacting pulses in a mathematically rigorous way. The equations describing the flow on the manifold is also given in an explicit form. By it, we can easily analyze the movement of pulses such as repulsiveness, attractivity and/or the existence of bound states of pulses. Interaction of front solutions are also treated in a similar way.

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

Abstract The renormalization group (RG) method as a powerful tool for reduction of evolution equations is formulated in terms of the notion of invariant manifolds. We start with derivation of an exact RG equation which is analogous to the Wilsonian RG equations in statistical physics and quantum field theory. It is clarified that the perturbative RG method constructs invariant manifolds successively as the initial value of evolution equations, thereby the meaning to set t0=t is naturally understood where t0 is the arbitrary initial time. We show that the integral constants in the unperturbative solution constitutes natural coordinates of the invariant manifold when the linear operator A in the evolution equation is semi-simple, i.e., diagonalizable; when A is not semi-simple and has a Jordan cell, a slight modification is necessary because the dimension of the invariant manifold is increased by the perturbation. The RG equation determines the slow motion of the would-be integral constants in the unperturbative solution on the invariant manifold. We present the mechanical procedure to construct the perturbative solutions hence the initial values with which the RG equation gives meaningful results. The underlying structure of the reduction by the RG method as formulated in the present work turns out to completely fit to the universal one elucidated by Kuramoto some years ago. We indicate that the reduction procedure of evolution equations has a good correspondence with the renormalization procedure in quantum field theory; the counter part of the universal structure of reduction elucidated by Kuramoto may be Polchinskis theorem for renormalizable field theories. We apply the method to interface dynamics such as kink–anti-kink and soliton–soliton interactions in the latter of which a linear operator having a Jordan-cell structure appears.

Pulse-pulse interaction in reaction-diffusion systems

Abstract It had been long believed that one-dimensional travelling pulses and the corresponding two-dimensional expanding rings and spiral waves arising in excitable reaction–diffusion systems annihilate when they closely approach one another. However, recently it has been numerically confirmed that if the velocity is very slow, expanding rings and spiral do not necessarily annihilate. In particular, in some situation, two closely approaching pulses reflect, as if they were elastic like objects. By using the center manifold theory, we show that if there are travelling pulses which primarily and super-critically bifurcate from a standing pulse when some parameter is varied, they possess reflection mechanism if the velocity is very slow.

Effect of domain-shape on coexistence problems in a competition-diffusion system

We discuss a competition-diffusion system to study coexistence problems of two competing species in a homogeneous environment. In particular, by using invariant manifold theory, effects of domain-shape are considered on this problem.

2n-splitting or edge-splitting? — A manner of splitting in dissipative systems — : To the memory of Masaya Yamaguti ()

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 Nishiura-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 1D-case, this means that the number of newly born pulses increases like 2k afterk-th splitting, not 2n-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 “2n-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.

Dynamics of interfaces in competition-diffusion systems

This paper is concerned with the dynamics of interfaces in the Lotka-Volterra competition-diffusion system\[ u_1 = \epsilon ^2 \Delta u + u( {1 - u - cw} ), \]\[ w_1 = \epsilon ^2 D\Delta w + w( {a - bu - w} ),\quad {\text{in }}R^n , \] where

Dynamics of metastable localized patterns and its application to the interaction of spike solutions for the Gierer-Meinhardt systems in two spatial dimensions

\epsilon > 0

Emergence of adaptability to time delay in bipedal locomotion

is a small parameter and

Self-motion of camphor discs. Model and analysis

D > 0

Dynamics of Front Solutions in a Specific Reaction-Diffusion System in One Dimension

is a constant. If