Trajectories of a Solitary Wave for the KdV Equation with Variable Coefficients

time steps

50

wavenumber k

2.

frequency

ω

1

1.5

frequency

ω

2

1.5

amplitude

Q

1

1.25

amplitude

Q

2

1.

initial position δ

0

function choice

1

2

3

One aspect of David Bohm's causal interpretation of quantum theory is that the formalism is applicable to all linear or nonlinear partial differential equations (PDEs or nPDEs) obeying the continuity equation. In this Demonstration the trajectory concept is applied to the Korteweg–de Vries equation with variable coefficients for a singular solitary wave-like solution. The time evolution of the soliton position

x(t)

could be interpreted as streamlines of the wave or as idealized test particles, which do not interact by themselves and do not influence the wave, carried by a wave that obeys the KdV equation. The system is time reversible:

t→-t

. The graphics show the wave density and the trajectories in

x-t

space for various choices of the function

f(t)

.