Dynamics of localized patterns in nonlinear evolutional systems

在本计画中，我们考虑反应扩散系统中的定域模式，并尝试建立分析其时间演化行为的理论。作为结果，我们得到了一些结果的系统在一维问题，如分歧结构和脉冲相互作用。在此基础上，推导了描述相互作用脉冲运动的方程，并给出了数学上的正确性。此外，将它们考虑在分叉点附近，应用标准中心流形理论和脉冲相互作用方法，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性图案的基础。

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.