The study of solitary waves or solitons started in the nineteenth century when a young Scottish engineer, John Scott-Russell, observed a single wave which, guided through a local canal, seemed to travel as far as he could see without losing its shape. He jumped on a horse, so the story goes, and followed the wave for miles along the canal. A soliton is just a single bump that does not broaden, lose its shape or weaken as it travels along in a medium (see figure).

The stability of solitons is truly remarkable, and this makes them very relevant for applications, varying from error-free transmission of light pulses through a glass fiber, or electric pulses through nerve cells, to understanding tsunamis.

For a long time there was a fierce debate involving many scientific celebrities about whether solitary waves could really exist; because of the natural tendency of waves to broaden and dissipate, the phenomenon required a miraculous conspiracy. The debate was settled when Korteweg and his student De Vries wrote down their equation for water waves in a shallow rectangular canal in 1895, and rigorously showed that the equation did indeed have solitary wave solutions. The equation is nonlinear because the function u appears quadratic in the last term, indeed this equation is very different from the linear equations for electromagnetic waves, yet it is the prototype of large families of nonlinear soliton equations which share similar properties.

Solitons have other remarkable properties, for example, their speed of propagation is proportional to their amplitude or height – so a big wave will overtake a small one. Furthermore, if solitons meet and interact, they will eventually re-emerge unaffected and continue their trip with a slight time delay. So these waves are not able to destroy each other, and that is a property which makes them extremely useful.