The Complex Korteweg-de Vries Equation: A Deeper Theory of Shallow Water Waves

TL;DR

This paper derives a complex Korteweg-de Vries equation for shallow water waves, providing a more fundamental framework that captures particle trajectories and wave dynamics more comprehensively than traditional real KdV models.

Contribution

It introduces a complex KdV equation derived from Levi-Civita's fluid theory, establishing its fundamental role over the real KdV in shallow water wave analysis.

Findings

Complex KdV describes all water particle motions from surface to bottom.

Wave elevation is governed by the real KdV as a natural consequence.

Results align well with existing numerical simulations.

Abstract

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.