Three-Soliton Collision in the Trajectory Approach

time steps

25

starting point 1

-20

starting point 2

-5

starting point 3

10

initial values

P

1

wavenumber

3

13

P

2

wavenumber

8

13

P

3

wavenumber

1

κ

1

constant

-5

κ

2

constant

0

κ

3

constant

-5

This Demonstration determines the streamlines or trajectories of idealized particles in a three-soliton collision, according to the Korteweg–de Vries equation (KdV) in

(x,t)

space. The collision of three solitons with different amplitudes involves the wave numbers

P

1

,

P

2

, and

P

3

. 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

κ

1

,

κ

2

, and

κ

3

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

(x,t)

space.