The emergence of Dynamic Mode Decomposition (DMD) as a practical way to
attempt a Koopman mode decomposition of a nonlinear PDE presents exciting
prospects for identifying invariant sets and slowly decaying transient
structures buried in the PDE dynamics. However, there are many subtleties in
connecting DMD to Koopman analysis and it remains unclear how realistic Koopman
analysis is for complex systems such as the Navier-Stokes equations. With this
as motivation, we present here a full Koopman decomposition for the velocity
field in Burgers equation by deriving explicit expressions for the Koopman
modes and eigenfunctions - the first time this has been done for a nonlinear
PDE. The decomposition highlights the fact that different observables can
require different subsets of Koopman eigenfunctions to express them and
presents a nice example where: (i) the Koopman modes are linearly dependent and
so cannot be fit a posteriori to snapshots of the flow without knowledge of the
Koopman eigenfunctions; and (ii) the Koopman eigenvalues are highly degenerate
which means that computed Koopman modes become initial-condition dependent. As
way of illustration, we discuss the form of the Koopman expansion with various
initial conditions and assess the capability of DMD to extract the decaying
nonlinear coherent structures in run-down simulations.

通过对 Burgers 方程中速度场进行全 Koopman 分解，我们首次得到了非线性 PDE 的 Koopman 模式和特征函数的显式表达式。该分解突出了不同观测量需要不同的 Koopman 特征函数来表示的事实，并评估了 DMD 提取衰减的非线性相干结构的能力。