The KdV equation can be written

, where

, and so on. The motion of the particles is governed by the current flow

, where

is the velocity, which is derived from the continuity equation

. The guiding equation is based only on the velocity term,

, with

. We get the starting points of possible trajectories inside the wave that are linearly distributed according to the wave density

and lead to single trajectories:

If the velocity could be expressed by

, then the total derivative

is simplified to

and it follows that the acceleration is

for the single particles in the wave. In this case, the acceleration is based only on the velocity term

, where the wave potential

has a similar form of the quantum potential of the causal interpretation of quantum mechanics. Only for the solitary wave

is unequal to zero, which cancels out the term proportional to

, which further leads to a zero acceleration. Keep in mind that, for the rational solution, the conservation of total mass

fails. Further investigation will be necessary to answer many open questions.

[1] A. D. Polyanin and V. F. Zaitsev,

*Handbook of Nonlinear Partial Differential Equations*, London: Chapman & Hall/CRC Press, 2003.