论文标题
迈向计算Morse-loer同源性：通过计算临界点的相对索引来迫使连接轨道的结果
Towards computational Morse-Floer homology: forcing results for connecting orbits by computing relative indices of critical points
论文作者
论文摘要
为了取得更好的Morse-loer同源性计算性，从而增强了浮动理论的适用性，必须拥有确定平衡相对指数的工具。由于即使存在非平凡的固定点也很难实现，因此提取其索引信息通常是遥不可及的。在本文中，我们建立了一种计算机辅助的证明方法来确定固定状态的相对指标。我们介绍了一般框架，然后专注于部分微分方程描述的三个示例问题，以显示这些想法在实践中的工作方式。基于严格的实现，并提供了随附的代码，我们确定了许多固定点的相对索引。此外，我们展示了如何使用强迫结果证明有关在偏微分方程中连接轨道和行进波的定理。
To make progress towards better computability of Morse-Floer homology, and thus enhance the applicability of Floer theory, it is essential to have tools to determine the relative index of equilibria. Since even the existence of nontrivial stationary points is often difficult to accomplish, extracting their index information is usually out of reach. In this paper we establish a computer-assisted proof approach to determining relative indices of stationary states. We introduce the general framework and then focus on three example problems described by partial differential equations to show how these ideas work in practice. Based on a rigorous implementation, with accompanying code made available, we determine the relative indices of many stationary points. Moreover, we show how forcing results can be then used to prove theorems about connecting orbits and traveling waves in partial differential equations.
