# Solitons: A Cowboy Story of Discovery (Sort Of)

Sometimes, mathematics that was developed to explain what may seem like a relatively simple physical system is in fact quite profound and crops up all over the place after it has been developed. This is the case with solitons – solitary waves that first piqued the interest of physicists 190 years ago and continue to be investigated in contemporary physics research.

The story goes that while riding on horseback beside a canal near Edinburgh in 1834, a Scottish engineer and shipbuilder named John Scott Russell saw a boat suddenly stop and watched the wake of the boat continue on ahead at the same speed and without deforming. He chased after it on his horse for several miles until he lost it ‘in the windings of the channel’, then went home and wrote a report about it to the British Science Association. He is often accredited with being first person to observe a soliton – or as he put it, ‘that singular and beautiful phenomenon which I have called the Wave of Translation’ – which of course is very unlikely to be the case. In truth, lots of people before him had likely seen solitary waves like the one shown at the top of this blog but weren’t inspired or enabled go in hot pursuit on horseback and then run home and write about it. However, because of his profession and expertise, Russell was the first person to both see the phenomenon and understand that it contradicted the predictions of physical theory at the time.

Before Russell’s canal-side horseback ride, the physical theory used to describe waves on the surface of water involved only linear equations. ‘Linear’ means that an equation is just made of things multiplied by some number and added together – linear equations are the simplest type of equation possible. With that said, linear equations describing waves on the surface of water have sine waves as a ‘solution’ (where solution in this context really just means the predicted behaviour). That’s a reasonable prediction for the mathematics to make when it comes to water waves – sine waves are in a sense the ‘quintessential wave’ and water surface disturbances often do look like them:

So, Russell’s observation of a water wave that wasn’t sinusoidal and didn’t just flatten out and disappear showed that the mathematics used to describe the physics of water waves didn’t tell the whole story. He set up experiments in a wave tank in his lab and managed to recreate the solitary waves in shallow channels (like a canal) and measured how the speed of the solitary waves depended on the depth of the water and the height of the wave’s peak. Although he didn’t manage to put together new equations that could actually predict the existence of his solitary waves, he didn’t have to work on it alone anymore because this stage his newly-observed phenomenon had ‘made a splash’ amongst the who’s who of nineteenth century British physics.

One notable character who took to working on the problem was George Stokes, a physicist and mathematician who was born in rural county Sligo and went on to study and work in Cambridge for his entire adult life. He is known among other things for his contribution to the development of the Navier-Stokes equation, an equation which describes the internal motion of a fluid and is so intricate that it has a one million dollar prize offered to anybody who can prove or disprove that smooth solutions always exist. The point is, if anyone could be expected to quickly put together a mathematical model to predict the presence of Russel’s solitary waves, it would be Stokes or contemporaries of his such as Airy. However, the problem alluded these talented physicists and mathematicians and it took over sixty years from Russell’s sighting of the solitary wave in the canal in Edinburgh for a satisfactory equation to describe shallow water waves and predict solitary waves to be put forward by a PhD student and supervisor called Diederik Johannes Korteweg and Gustav de Vries in 1895. Hence, the equation is known as the KdV equation.

Once the problem had been resolved, everybody lost interest and nobody did anything much with the KdV equation – after all, it only described water waves in shallow water, and new, cool quantum physics and relativity was the talk of the physics town. Until 1960 that is, when the new physics repeatedly spat out the same KdV equation – at first it fell out of magnetohydrodynamics (the study of electrically conducting fluids such as those inside the sun or a nuclear fusion reactor), then it showed up in other places, like optics, plasma physics, lattice dynamics, and material physics. Knowing the details of each of these fields of physics is not too important here – what I want to focus on is that in a wide variety of topics where waves could show up, the KdV equation has shown up and where the KdV equation exists, solitary wave solutions exist.

It’s at this point that the characteristic of Russell’s solitary waves that excited him so much becomes more interesting. Russell was amazed at the persistence of the wave’s form in staying unchanged over the miles that he followed it and although that may not be so interesting to you and I when considering the surface of water, it means that in all of these other complex physical systems where the KdV equation has shown up, persistently stable solitary waves can exist and move about. In the 1960s, it was observed that the stability of the solitary wave solutions made them behave like particles. Hence, the first half of ‘solitary’ was contracted with the last half of ‘proton’, ‘neutron’, or ‘electron’ to rename Russel’s ‘solitary Waves of Translation’ as ‘solitons’. Solitons continue to be researched throughout mathematical physics and present new avenues to physics discovery even now, 190 years after Russel’s horseback ride.

References:

[1]: Mark J. Ablowitz and Peter A. Clarkson. Solitons, Nonlinear Evolution Equations and Inverse Scattering. Vol. 149. Cambridge University Press, 1991.

[2]: John S. Russell. Report on Waves: Made to the Meetings of the British Association in 1842-43. Richard and John E. Taylor, 1845.

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