论文标题
非局部系统中的相干结构 - 功能分析工具
Coherent structures in nonlocal systems -- functional analytic tools
论文作者
论文摘要
我们开发用于分析具有非本地耦合的空间扩展系统中的前部，脉冲和波列的工具。我们首先确定线性操作员的弗雷霍尔姆性质，从而确定主要部分的重点可逆性，并根据必要和充分的条件在空间无穷大处的可逆性。然后，我们建立在弗雷德·霍尔姆（Fredholm）理论的基础上，以在最佳规则性假设下构建非局部空间动力学的中心歧管，而矢量场和相空间减少了，并且相位空间通过有界溶液的转移确定了后验。作为应用程序，我们仅使用非线性的$ c^1 $ regumentity在Lyapunov Center定理中建立了小型周期性波列的唯一性。
We develop tools for the analysis of fronts, pulses, and wave trains in spatially extended systems with nonlocal coupling. We first determine Fredholm properties of linear operators, thereby identifying pointwise invertibility of the principal part together with invertibility at spatial infinity as necessary and sufficient conditions. We then build on the Fredholm theory to construct center manifolds for nonlocal spatial dynamics under optimal regularity assumptions, with reduced vector fields and phase space identified a posteriori through the shift on bounded solutions. As an application, we establish uniqueness of small periodic wave trains in a Lyapunov center theorem using only $C^1$-regularity of the nonlinearity.
