  We investigate the spatiotemporal structure of simulations of the homogeneous
slab and isothermal plane models for the vertical motion in the Galactic disc.
We use Dynamic Mode Decomposition (DMD) to compute eigenfunctions of the
simulated distribution functions for both models, referred to as DMD modes. In
the case of the homogeneous slab, we compare the DMD modes to the analytic
normal modes of the system to evaluate the feasibility of DMD in collisionless
self gravitating systems. This is followed by the isothermal plane model, where
we focus on the effect of self gravity on phase mixing. We compute DMD modes of
the system for varying relative dominance of mutual interaction and external
potential, so as to study the corresponding variance in mode structure and
lifetime. We find that there is a regime of relative dominance, at
approximately $ 4:1 $ external potential to mutual interaction where the DMD
modes are spirals in the $ (z,v_z) $ plane, and are nearly un-damped. This
leads to the proposition that a system undergoing phase mixing in the presence
of weak to moderate self gravity can have persisting spiral structure in the
form of such modes. We then conclude with the conjecture that such a mechanism
may be at work in the phase space spirals observed in Gaia Data Release 2, and
that studying more complex simulations with DMD may aid in understanding both
the timing and form of the perturbation that lead to the observed spirals.
