Solitons
Solitons are solitary nonlinear pulses that can propagate over a
long distance without appreciable deformation. In dispersive wave media
in which the phase and group velocities differ from each other, any linear
waves are bound to deform through dispersion. One such example is shown
in http://physics.usask.ca/~hirose/ep225/disp.htm
in this Web site. Amazingly, dispersion can be compensated by nonlinearity
and a pulse of sufficiently large amplitude can propagate without deformation.
The first experimental observation of solitons was made by Scott Russel
in 1844 when he was experimenting water wave propagation along a canal.
See Solitons in Action, Ed. K. Lonngren and A. Scott, (Academic
Press, New York, 1978).
Animations shown here are based on exact solutions found by R. Miura
for solitons in water (and ion acoustic solitons in a plasma which can be described
by the same equation called Kortwegde Vries (KdV) equation).
Observe the following:

The propagation velocity of solitons depends on their amplitude. The velocity
increases with the amplitude and in the cases shown, both the velocity
and amplitude ratios are 4 to 1.

The pulse width decreases as the amplitude increases.

Collision does not destroy solitons as shown for passing collision. Both
solitons remain intact after collision.

At the instant of collision shown in a snapshot at t = 0, the total amplitude
becomes smaller. This is in quite contrast to linear waves for which the
superposition principle universally works. It does not at all for nonlinear
waves.
Two independent solitons. Amplitude ratio is 8:2 = 4:1 and so is
the velocity ratio.
Passing collision of the solitons shown above. Note that at the
instant of collision, the amplitude is only 6. However, the total area
associated with the pulses is constant.
Snapshots at t = 6, 0, and +4.
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