As you may read in the page about nonlinear science, the concern of nonlinear physics is often the unexpected appearance of chaos or order. Within this framework the soliton plays the role of order.

To explain the nature of solitons, we will consider the behaviour of water waves on shallow water. The scenario could be set e.g. in one of the canals, which was the 19'th century's analogue to nowaday's highways. Indeed, it was in such a location that a soliton was first noticed in 1849 by the Scottish engineer John Scott Russell. Imagine that a wave is somehow initiated in such a canal. One would expect that the wave then rolls along the canal while it spreads out and soon ends it's life as small wiggles on the surface. However, if certain conditions are fulfilled, the unexpected may happen; a soliton can be excited, and the wave will continue to roll along the canal without changing shape.

It turns out that a soliton is very robust against perturbations. The bottom of the canal may be uneven and bumpy; ducks and dogs may swim around in the canal; but the soliton will gently pass these obstacles.

The water wave soliton is a result of a dynamic balance between dispersion, i.e. the wave's tendency to spread out, and nonlinear effects. In order to substantiate this statement, we have to pass to some mathematics.

Balance of dispersion and nonlinearity

The dynamics of water waves in shallow water is described mathematically by the Korteveg - de Vries (KdV) equation:

ut +  uxxx +  u ux =  0,

u=u(x,t) measures the elevation at time t and position x, i.e. the height of the water above the equilibrium level. The subscripts denote partial differentiation. The second and the third term in the equation is the dispersive and the nonlinear term, respectively.

Dispersion

Let us first investigate the effect of the dispersive term. Thus we neglect the nonlinear term in the KdV equation. This leaves us with the following:

ut +  uxxx =  0.

dispersion  
 Click the button on the left hand side and you will see how the water wave will act when the dynamics is governed by the equation just above. A physicist trained in wave dynamics will know that waves with different wavelengths travel with different velocities. Since the initial wave is composed by many small waves with different wavelengths, it will soon spread out in the many components and can no longer be described as an entity or object.

Nonlinearity

Now let us see the effect of the nonlinear term. We neglect the dispersive term in the KdV equation, which leaves us with the following:

ut  +   u ux  =  0.

nonlinearity  
 Click the button on the left hand side and you will see how the water wave will act when the dynamics is governed by the nonlinear equation just above. Clearly, the title of this animation should be "Breaking the waves". The top of the wave moves faster than the low sides and this causes the wave to shock in the same way as the waves we see on the beach.

The soliton

When both the dispersive and the nonlinear term are present in the equation the two effects can neutralize each other. If the water wave has a special shape the effects are exactly counterbalanced and the wave rolls along undistorted. The soliton shape can be found by direct integration of the KdV equation:

u(x,t)  =   a sech2[b(x-vt)],

with b=(a/12)1/2 and v=3a. The constant a is the only free parameter in the solution. It defines the amplitude and the width in such a way that a large (tall) soliton will be narrow, while a low soliton will be broad. The constant v defines the velocity of the soliton. Since v=3a a tall soliton will move faster than a low one.
soliton  
 Click the button to the left in order to see an example of a soliton with a=1.

Soliton dynamics

The concept of a soliton provides a new line in research in systems which are wave-like in nature, e.g. systems involving water or light. The soliton is regarded as an entity, a quasi-particle which conserves it's character and interacts with the surroundings and other solitons as a particle. In stead of trying to describe the complete field of water or light, one can concentrate on the evolution of solitons which will stay around in the system (in principle) forever and will constitute the most important part of the dynamics.

We give here just a single example of soliton dynamics.

Soliton collision

collision  
 Click the button on the left hand side to see a collision of two solitons. The tall, fast soliton overtakes the broad, slow one. Remark that the collision does not result in any small waves or wiggles after the interaction. The two solitons look exactly as they did before the collision. This particle-like behavior is generic for the soliton. Therefore a phenomenological definition of the soliton could be a localized wave that does not radiate (generate small waves) during collisions with other solitons.

This is of course not the complete story about solitons. There is more out there:

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