论文标题
周期性Korteweg-de Vries方程的Koopman分析
Koopman analysis of the periodic Korteweg-de Vries equation
论文作者
论文摘要
Koopman操作员的特征光谱使非线性动力学分解为状态空间的非线性函数，并具有纯粹指数和正弦的时间依赖性。对于有限数量的动态系统，可以通过分析和分析找到这些Koopman征函数。在这里，使用周期性的反向散射变换和一些代数几何概念，这是针对Korteweg-de Vries方程进行的。据作者所知，这是对没有微不足道的全球吸引子的偏微分方程的第一个完整的Koopman分析。结果显示，通过数据驱动的动态模式分解方法（DMD）计算出的频率。我们证明，通常DMD在假想轴附近给出了大量的特征值，并在这种情况下表明应如何解释这些特征值。
The eigenspectrum of the Koopman operator enables the decomposition of nonlinear dynamics into a sum of nonlinear functions of the state space with purely exponential and sinusoidal time dependence. For a limited number of dynamical systems, it is possible to find these Koopman eigenfunctions exactly and analytically. Here, this is done for the Korteweg-de Vries equation on a periodic interval, using the periodic inverse scattering transform and some concepts of algebraic geometry. To the authors' knowledge, this is the first complete Koopman analysis of a partial differential equation which does not have a trivial global attractor. The results are shown to match the frequencies computed by the data-driven method of dynamic mode decomposition (DMD). We demonstrate that in general DMD gives a large number of eigenvalues near the imaginary axis, and show how these should be interpretted in this setting.
