Three-Soliton Collision in the Trajectory Approach
This Demonstration determines the streamlines or trajectories of idealized particles in a three-soliton collision, according to the Korteweg–de Vries equation (KdV) in space. The collision of three solitons with different amplitudes involves the wave numbers , , and . These are, in turn, determined by the dispersion relations, given the speed of each wave. For the three-soliton system, the wave velocity depends on the amplitude. The constants , , and determine the initial positions of the peaks of each soliton. The streamlines of the particles follow the current flow, which can be derived from the continuity equation. The concept of a trajectory is based on the causal interpretation of quantum mechanics developed by David Bohm. The three-soliton result is obtained by the Hirota direct method. The graphic on the left shows the density (blue) and the velocity (green) of the idealized particles. On the right, you can see the density and the trajectories in space.
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
, , ,
, , and .
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.
 R. Hirota, "Exact Solution of the Korteweg–de Vries Equation for Multiple Collisions of Solitons," Physical ReviewLetters, 27(18), 1971 pp. 1192–1194. doi:10.1103/PhysRevLett.27.1192.