Soliton Trajectories According to Bohmian Quantum Mechanics

wave numbers

p1

p2

This Demonstration presents the motion of idealized particles inside a two-soliton using the Korteweg–de Vries equation (KdV):

∂

t

u+6u∂

x

u+∂

x,x,x

u=0

in

(x,t)

space. The interaction of a two-soliton depends on the wave numbers p1 and p2 that are related via the dispersion relation to the speed of the each wave. In this case

p1

2

and

p2

2

are the velocities of the two solitary waves. The motion of the particles is governed by the current flow, which is derived from the continuity equation

∂

t

u+∂

x

(vu)=0

directly. The guidance equations are based only on the velocity, which is

v=

∂

x,x

u

+3u

. With

u(x,t=0)=x

i

(t

0

)

we get the starting points of possible trajectories inside the wave that are distributed according to the density of the wave and lead to single trajectories:

x(t)=∫

t

0

v(x,t')t'

. For the calculation an initial Gaussian distribution is chosen. The system is time reversible:

t-t

.The concept is based on the causal interpretation of quantum mechanics developed by David Bohm.