In 1971 Hirota introduced a new direct method for constructing multi-soliton solutions to the integrable nonlinear Korteweg–de Vries equation

with the partial derivative

and so on. The Hirota direct method makes a transformation to a bilinear equation via the transformation

so that in the new form, the multi-soliton solutions can be solved using the Hirota

-operator.

Specifically, the bilinear equation for the Korteweg–de Vries equation is

,

which can be solved by the Hirota

-operator

.

In the original work of Hirota, the KdV equation is solved using the ansatz

,

where the

matrix

has the form

,

where

is the Kronecker delta and

with the constant phase factor

.

In practice, for the three-soliton, the

matrix

is given by

,

The motion of the particles associated with the current flow can be extracted from the continuity equation

. The guiding equations depend only on the velocity, which for the KdV equation become

.

With the transformation

the velocity term, which is also bilinear, is

.

The starting points of possible trajectories inside the wave can be chosen by the settings

,

, and

. The initial points should be distributed around the peak of each wave. The single trajectories are calculated using

. The system is time reversible:

. Due to the singularities, the size of the velocity term, and computational limitations, the wave numbers

have to be chosen carefully.

[1] R. Hirota, "Exact Solution of the Korteweg–de Vries Equation for Multiple Collisions of Solitons,"

*Physical Review* *Letters*,

**27**(18), 1971 pp. 1192–1194.

doi:10.1103/PhysRevLett.27.1192.