Complex Korteweg–de Vries equation: A deeper theory of shallow water waves

Using Levi-Cività'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 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.

Figure 1

Schematic of water layer with average depth $h$. The function $\eta (x,t)$ describes the water elevation above the average level.

Figure 2

An illustration of the paths of the particles in a rightward-propagating cnoidal wave (not to scale). The surface elevation is described by the real KdV equation, but the motion of the particles within the fluid is described by the complex KdV equation (16). The motion of the particle at a given point is determined by (30) and (31). The trajectory depends on the wave modulus, frequency, and depth in the fluid.