  Using Levi-Civita's theory of ideal fluids, we derive the complex Korteweg-de
Vries (KdV) equation, describing the complex velocity of a shallow fluid up to
first order. We use perturbation theory, and the long wave, slowly varying
velocity approximations for shallow water. The complex KdV equation describes
the nontrivial dynamics of all water particles from the surface to the bottom
of the water layer. A crucial new step made in our work is the proof that a
natural consequence of the complex KdV theory is that the wave elevation is
described by the real KdV equation. The complex KdV approach in the theory of
shallow fluids is thus more fundamental than the one based on the real KdV
equation. We demonstrate how it allows direct calculation of the particle
trajectories at any point of the fluid, and that these results agree well with
numerical simulations of other authors.
