Dynamics of patterns and their interactions for nonlinear evolutional equations

在本项目期间，我们对一维行波解的分叉结构进行了完整的分类，并从动力系统的角度分析了非均匀介质中的行波解的动力学。通过分析，我们可以用不变流形理论的话知道解是如何穿过或反映在障碍物上的。文中还讨论了两个前沿解之间的相互作用。一般情况下，脉冲液的治疗是困难的。利用两个前沿解相互作用的结果，我们将脉冲解构造为两个前沿解的组合，从而使脉冲解的处理变得更加容易。第二个主要结果是建立了一个分析二维域中边界尖峰解的一般框架。我们证明了尖峰解沿边界向边界曲率最大的点移动。我们还证明了在具有最大曲率的边界点的邻域内存在稳定的双峰驻定解。将结果应用于Gierer-Meinhardt模型及其影子系统。这些结果在高维空间的推广仍然是开放的。对于包括螺旋运动在内的二维反应扩散系统的界面动力学问题，目前还没有完成严格的处理。但为了研究这种界面问题，几乎所有必要的准备工作都已经完成，如可视化软件和螺旋芯运动搜索程序等。在项目之初，我们除了模糊的期望外，并不知道解决这种界面问题的明确方向。在这个项目的最后阶段，现在我们有了明确的方法来实现目标。项目期间取得的成果将作为不可或缺的重要成果传给下一个项目。

In the period of this project, we completely classified the bifurcation structures of 1 dimensional traveling front solutions and analyzed the dynamics of traveling front solutions in inhomogeneous media from the dynamical system point of view. By the analysis, we can know how the solutions go through or reflect by obstacles with words of invariant manifold theory. The interaction of two front solutions is also treated. In general, the treatment of pulse solution is difficult. Applying the results of the interaction of two front solutions, we construct a pulse solution as the combination of two front solutions, which makes easier to treat pulse solutions. Second main result is to establish a general framework to analyze the boundary spike solutions in two dimensional domain. We proved a spike solution moves along the boundary towards the point with maximal curvature of the boundary. We also showed the existence of stable stationary solutions with two peaks in the neighborhood of a point on the boundary with maximal curvature. The results are applied to the Gierer-Meinhardt model and its shadow system. The extension of these results to higher dimensional spaces is still open. For the interfacial dynamics of reaction-diffusion systems in two dimensional spaces including spiral motion, the rigorous treatment has not been completed yet. But almost any necessary preparation in order to study such an interfacial problem such as visualization soft ware and the search program of the motion of a spiral core has been done. At the beginning of the project, we did not know explicit direction to solve this kind of interfacial problems except ambiguous expectations. At the end of the period of this project, now we have explicit ways to reach the goal. The results obtained during the period of the project are inherited to the next project as indispensable and important results.