A Koopman decomposition of a complex system leads to a representation in which nonlinear dynamics appear to be linear. The existence of a linear framework with which to analyze nonlinear dynamical systems brings new strategies for prediction and control, while the approach is straightforward to apply to large datasets using dynamic mode decomposition (DMD). However, it can be challenging to connect the output of DMD to a Koopman analysis since there are relatively few analytical results available, while the DMD algorithm itself is known to struggle in situations involving the propagation of a localized structure through the domain. Motivated by these issues, we derive a series of Koopman decompositions for localized, finite-amplitude solutions of classical nonlinear PDEs for which transformations to linear systems exist. We demonstrate that nonlinear traveling wave solutions to both the Burgers and KdV equations have two Koopman decompositions; one of which converges upstream and another which converges downstream of the soliton or front. These results are shown to generalize to the interaction of multiple solitons in the KdV equation. The existence of multiple expansions in space and time has a critical impact on the ability of DMD to extract Koopman eigenvalues and modes which must be performed within a temporally and spatially localized window to correctly identify the separate expansions. We provide evidence that these features may be generic for isolated nonlinear structures by applying DMD to a moving breather solution of the sine-Gordon equation.