A soliton wave (or solitary wave) is an unusual phenomenon in wave mechanics, in which the nonlinearity of the medium's resistance to the wave, which tends to make it narrower, exactly matches the dispersion of the medium, which tends to make it wider.
In mathematics, one type of soliton wave is described by the solution to the Korteweg-de Vries (KdV) equation:
ut + u ux + uxxx = 0
where:
ut(x, t) = ∂u(x, t) / ∂t
ux(x, t) = ∂u(x, t) / ∂x
uxxx(x, t) = ∂3u(x, t) / ∂x3
For those of you with browsers that don't support the character, ∂ is the partial differential symbol.
uxxx is the dispersive term, which tends to delocalise the wave. u ux is the nonlinear term, which tends to concentrate the wave. By integrating the equation, a solution is found:
u(x, t) = a sech2(b(x - vt))
where:
b = (a / 12)1/2
v = 3 a
a is the only free variable, and describes the amplitude of the soliton. The equation shows that tall solitons are narrow, and short ones are wide. v is the velocity of the wave, and as v = 3 a, taller solitons go faster.
The most obvious of the unusual properties of a soliton is its longevity. A soliton can travel a long way without losing shape or much amplitude. A less obvious but more intriguing property is its identity… if two solitons are travelling on the same medium and collide, rather than merging the characteristics of the two waves, they seperate again almost unchanged after they hit each other!
John Scott Russell was the first to observe and characterise the soliton, in a canal in 1834, and in his writings he referred to it as the Wave of Translation.
Sources:
http://www.usf.uni-osnabrueck.de/~kbrauer/solitons.html
http://www.imm.dtu.dk/math_phys/Solitons.html