We consider Koopman operator analysis for partial differential equations (PDEs) describing spatially-extended systems. For PDEs with a stable stationary solution, we can formally introduce Koopman eigenfunctionals of the system and use them to define the isostables, which characterize asymptotic convergence of the system to the stationary state. It is shown that for a few special PDEs, i.e., diffusion, Burgers, and nonlinear phase-diffusion equations, the Koopman eigenfunctionals can be obtained analytically. We also consider reaction-diffusion PDEs with a stable limit-cycle solution, which describe rhythmic spatiotemporal patterns. In this case, the Koopman eigenfuncionals can be used to define the isochrons and isostables of the system. It is shown that linear approximation of the Koopman eigenfunctionals around the limit-cycle solution yields reduced low-dimensional phase-amplitude equations, which can be used to analyze synchronization dynamics of the rhythmic spatiotemporal patterns.
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