Dynamics of localized patterns in nonlinear evolutional systems

在本项目中，我们考虑了反应扩散系统的局部模式，并试图建立理论来分析时间演化行为。得到了系统在一维问题中的分支结构和脉冲相互作用等结果。明确地说，类脉冲局域模式的尾部呈指数衰减，然后推导了描述相互作用脉冲运动的方程及其数学有效性。此外，通过在分岔点的邻域中考虑它们，应用标准中心流形理论和脉冲相互作用方法，我们可以分析脉冲的各种复杂行为，如脉冲的自复制和反射。例如，在自我复制现象中，我们可以证明临界距离的存在，当脉冲之间的距离超过临界距离时，脉冲开始分裂。它有望解释空间周期模式的基础。

In this project, we considered the localized patterns in reaction-diffusion systems and tried to establish the theories to analyze the time evolutional behaviors. As the consequence, we obtained several results for the systems in 1D problems such as the bifurcation structures and pulse interactions. Explicitly speaking, he tails of pulse-like localized patterns are exponentially decaying, then we derived the equations describing the motion of interacting pulses as well as the mathematical validity. Moreover, by considering them in the neighborhood of bifurcation points and applying the standard center manifold theory together with pulse interaction methods, we could analyze various complex behaviors of pulses such as self-replication and reflection of pulses. For example, in self-replicating phenomena, we could show the existence of critical distances such that pulses begin to split when distances between pulses are beyond the critical one. It is expected to explain the basis of spatially periodic patterns.