Evaluate
WOLFRAM|DEMONSTRATIONS PROJECT

Soliton Trajectories According to Bohmian Quantum Mechanics

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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
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.