Koopman analysis of isolated fronts and solitons

Abstract

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 analyse nonlinear dynamical systems brings new strategies for prediction and control, while the approach is straight-forward to apply to large datasets owing to a connection with 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 localised structure through the domain. Motivated by these issues, we derive a series of Koopman decompositions for localised, finite-amplitude solutions of classical nonlinear PDEs. We first demonstrate that nonlinear travelling wave solutions to both the Burgers and KdV equations have two Koopman decompositions; one of which converges upstream and another which converges the other downstream of the soliton or front. We then use the inverse scattering transform to derive a full Koopman decomposition for (pure soliton) solutions to the KdV equation, identifying Koopman eigenvalues, eigenfunctions and modes. Our analysis indicates that there are many possible Koopman decompositions when the solution involves the interaction of multiple solitons. 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 localised window to correctly identify the separate expansions. In addition, 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.