a soliton
/ SOL-i-ton /
A soliton is a wave that refuses to fall apart. Throw a stone in a pond and the ripples spread and fade; but in just the right nonlinear medium, a single hump can travel for enormous distances holding its exact shape, and even survive a head-on collision with another such hump as if nothing happened. This particle-like wave is the soliton, and it is the signature phenomenon of integrable nonlinear PDEs — the thing that makes the field beautiful.
Here is why it exists. In a dispersive medium, different wavelengths travel at different speeds, so any localized pulse normally spreads out and flattens. Separately, a nonlinearity tends to make taller parts move faster, which steepens and would break the wave. A soliton is the exact compromise where these two destructive tendencies cancel: the spreading from dispersion is precisely undone by the focusing from nonlinearity, and the result is a permanent, self-reinforcing travelling shape. The three classic examples each realize this balance differently: the KdV soliton (a sech^2 hump in shallow water), the NLS bright soliton (a sech-shaped light pulse), and the sine-Gordon kink (a travelling twist). 'Solitary wave' is the looser term for any such localized travelling solution; 'soliton' is usually reserved for the stronger case where the waves also survive collisions unchanged — a property tied to the equation being integrable.
Why does it matter? Solitons overturned the intuition that nonlinearity always destroys coherence: here nonlinearity, far from spoiling the wave, is precisely what STABILIZES it. They are real and useful — optical solitons carry data through fibres, and solitons appear in plasmas, Bose-Einstein condensates, DNA models, and tsunamis. The honest caveat: the cleanest soliton behaviour (perfect collisions, infinitely many conserved quantities) belongs to integrable equations, which are special. Many physical systems have solitary waves that are robust but not exactly integrable, so they may radiate a little energy or wobble after collisions.
The KdV soliton u = (c/2) sech^2((sqrt(c)/2)(x - c t)) is the textbook case: a single hump whose height fixes its speed, and two of them pass through each other intact. Contrast an ordinary wave packet in a linear dispersive medium, which inexorably spreads and lowers as it travels — the soliton's permanence is the nonlinearity's gift.
A travelling hump that keeps its shape and survives collisions.
'Solitary wave' and 'soliton' are not quite synonyms: a solitary wave is just a localized travelling solution; a soliton additionally survives collisions unchanged, which is a hallmark of integrable equations. Many real-world 'solitons' are robust solitary waves that are not exactly integrable.