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Various motions of camphor boats in the water channel exhibit both a homogeneous and an inhomogeneous state, depending on the number of boats, when unidirectional motion along an annular water channel can be observed even with only one camphor boat. In a theoretical research, the unidirectional motion is represented by a traveling wave solution in a model. Since the experimental results described above are thought of as a kind of bifurcation phenomena, we would like to investigate a linearized eigenvalue problem in order to prove the destabilization of a traveling wave solution. However, the eigenvalue problem is too difficult to analyze even if the number of camphor boats is 2. Hence we need to make a reduction on the model. In the present paper, we apply the center manifold theory and reduce the model to an ordinary differential system.
The motion of a transition layer for a bistable reaction diffusion equation with heterogeneous environment
In this paper we study the dynamics of a single transition layer of a solution to a spatially inhomogeneous bistable reaction diffusion equation in one space dimension. The spatial inhomogeneity is given by a function $a(x)$. In particular, we consider the case where $a(x)$ is identically zero on an interval $I$ and study the dynamics of the transition layer on $I$. In this case the dynamics of the transition layer on $I$ becomes so-called very slow dynamics. In order to analyze such a dynamics, we construct an attractive local invariant manifold giving the dynamics of the transition layer and we derive an equation describing the flow on the manifold. We also give applications of our results to two well known nonlinearities of bistable type.
We consider pulse-like localized solutions for reaction-diffusion systems on a half line and impose various boundary conditions at one end of it. It is shown that the movement of a pulse solution with the homogeneous Neumann boundary condition is completely opposite from that with the Dirichlet boundary condition. As general cases, Robin type boundary conditions are also considered. Introducing one parameter connecting the Neumann and the Dirichlet boundary conditions, we clarify the transition of motions of solutions with respect to boundary conditions.
Our aim in this paper is to prove the instability of multi-spot patterns in a shadow system, which is obtained as a limiting system of a reaction-diffusion model as one of the diffusion coefficients goes to infinity. Instead of investigating each eigenfunction for a linearized operator, we characterize the eigenspace spanned by unstable eigenfunctions.
In the present paper, a model describing the self-motion of a camphor disc on water is proposed. The stability of a standing camphor disc is investigated by analyzing the model equation, and a pitchfork type bifurcation diagram of a traveling spot is shown. Multiple camphor discs are also treated by the model equations, and the repulsive interaction of spots is discussed.
Two types of aggregation systems with Fisher-KPP growth are proposed. One is described by a normal reaction-diffusion system, and the other is described by a cross-diffusion system. If the growth effect is dominant, a spatially constant equilibrium solution is stable. When the growth effect becomes weaker and the aggregation effect become dominant, the solution is destabilized so that spatially non-constant equilibrium solutions, which exhibit Turing's patterns, appear. When the growth effect weakens further, the spatially non-constant equilibrium solutions are destabilized through Hopf bifurcation, so that oscillatory Turing's patterns appear. Finally, when the growth effect is extremely weak, there appear spatio-temporal periodic solutions exhibiting infinite dimensional relaxation oscillation.
The Gierer-Meinhardt system is a mathematical model describing the process of hydra regeneration. The authors of  showed that if an initial value is close to a spiky pattern and its peak is far away from the boundary, the solution of the shadow Gierer-Meinhardt system, called a interior spike solution, moves towards a point on boundary which is the closest to the peak. However it has not been studied how a solution close to a spiky pattern with the peak on the boundary, called a boundary spike solution moves along the boundary. In this paper, we consider the shadow Gierer-Meinhardt system and dynamics of a boundary spike solution. Our results state that a boundary spike moves towards a critical point of the curvature of the boundary and approaches a stable stationary solution.
We consider the mixed ODE-PDE system called a hybrid system, in which the two interfaces interact with each other through a continuous medium and their equations of motion are derived in a weak interaction framework. We study the bifurcation property of the resulting hybrid system and construct an unstable standing pulse solution, which plays the role of a separator for dynamic transition from standing breather to annihilation behavior between two interfaces.
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