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 Kortweg-de Vries (KdV) equation). Observe the following:

1. 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.
2. The pulse width decreases as the amplitude increases.
3. Collision does not destroy solitons as shown for passing collision. Both solitons remain intact after collision.
4. 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.