As unmathematical as possible: an important application of my research area explained with five videos and zero equations.

If you throw a pebble in a pond and look carefully at the waves it creates, you can see some interesting things. You’ll observe, for example, that waves with a long wavelength travel faster than those with a short wavelength. No matter what the initial shape of the wave front was, after a while you will see a sequence of circular waves that are shorter on the inside and longer on the outside. This effect is called dispersion.

The wake of a boat provides another example of dispersion.

A second effect that you can observe in the videos above is that the individual wave fronts do not travel at the same speed as the big pattern. The velocity of the individual crests is called the phase velocity, the velocity of the whole pattern is called the group velocity. Phase velocity (red) and group velocity (green). (Source: http://commons.wikimedia.org/wiki/File:Wave_group.gif)

These examples show typical behavior of waves. The wave fronts change shape and are torn apart. What I am interested in are the exceptions, the rare cases where waves do preserve their shape.

Waves in a narrow and shallow channel like in this video are described by the Kortweg-de Vries equation, not quite correctly named after the Dutch mathematicians Diederik Korteweg and Gustav de Vries. This equation is one of the most important examples of an integrable system. There are many related concepts that carry this name and they all are too complicated to explain here mathematically. However, the intuition behind these concepts is always the same, and easy to understand. A system is integrable if it has an unusually large amount of conserved quantities.

Like its name suggests, a conserved quantity is something that does not change in time. Most systems in physics have conserve quantities like energy and (angular) momentum, but not many more than those. One way to view a conserved quantity is to consider it as an extra constraint that the solution must satisfy. So if there are many conserved quantities, there are many constraints to be fulfilled. This helps to solve (integrate) the system, hence the name integrable system.

The Kortweg-de Vries equation has an infinite amount of conserved quantities. Therefore, in some sense, there are infinitely many constraints that the solution must satisfy. As a consequence, the wave does not have any freedom left to change its shape. Such a wave is called a soliton, because it occurs by itself and (at least to theoretical physicists) looks like a particle.

When two solitons collide, weird stuff happens. As to be expected their interaction is complicated and looks chaotic. The waves break apart and new crests are created. But then, magic!

After the interaction the two original waves appear again, as if nothing at all has happened!

If you want to know more about the kind of mathematics used to describe integrable systems, you can have a look at my article about geometric mechanics. If that’s all too theoretical for you, you can head to Gloucestershire in England to try and surf a soliton:

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