On a bright August day in 1834, John Scott Russell, a Scottish engineer, became acquainted with a wave which would forever be associated with him within the field of fluid dynamics and beyond. He was following a boat which only just fit into the canal it was travelling down.

Suddenly the boat stopped. From the bow of the ship, a lump of water swelled and burst forward. On horseback, Scott Russell continued on, and became increasingly intrigued as the lump seemed to travel without changing shape, sustaining itself down the channel.

In this way they travelled, man and wave, at a pace of about nine miles per hour. They travelled for almost two miles before the wave disappeared among the twists and turns of the canal. This wave, standing about 40cm in height and extending about nine metres along the canal, never left Scott Russell’s thoughts. In September 1844 he gave a report on his scientific investigations into what we now call the *solitary wave*.

## Just smile and wave boys, smile and wave…

The solitary wave is a *surface* wave: one that occurs at the fluid–air interface. Rudimentary surface wave theory, and usual intuition, predicts that surface waves disperse, rather than keeping their form as the solitary wave did.

Think of the last time you splashed around in a swimming pool. If you put your hands at the water level and push, a lump of water would roll off your hands. But unlike the solitary wave, it would disperse into a bunch of ripples rather than retaining its shape, stopping you from launching long-ranged splash attacks.

Mathematically, dispersion describes how the velocity of a sinusoidal wave depends on its wavelength. For simplicity, consider a fluid interface where the surface height only varies in one direction, say the $x$-direction. Typically water waves aren’t perfectly sinusoidal, but we can build up any surface profile from sine waves. The system then evolves by the sine waves of varying wavelengths travelling independently at the velocity prescribed by the dispersion relation.

## The times, and space, they are a-changin’

Dynamical systems where the time evolution depends on spatial variation are described by *partial differential equations*. These are equations involving both time derivatives and spatial derivatives. When there are multiple variables in play, a derivative with respect to only one of the variables is known as a partial derivative. For example, $\partial u/\partial x$, also written $u_{x}$, denotes the spatial derivative of the function $u(x,t)$.

Many physical phenomena are described by such equations, such as diffusion (by the diffusion equation), motion in fluids (by the Navier–Stokes equations), and gravity in the setting of general relativity (by Einstein’s field equations).

Linear PDEs, where linear means we do not have products of derivatives, tend to be easier to solve, as you can add two known solutions to get another. Complicated starting configurations can be built out of the sum of many simpler solutions, like when we built an arbitrary wave profile from many sine waves.

On the other hand, nonlinear PDEs cannot be solved with this strategy, and are solved mostly on an ad-hoc basis. Solitons, as they were historically understood, are particular solutions to some special nonlinear PDEs.

## KdV solitons

The KdV equation is

\[u_t-6uu_x + u_{xxx} = 0.\]

If we look for solutions which are travelling waves, that is to say we try the form $u(x,t) = f(x-ct)$, we can find a family of solutions of lumps which are small outside of a small region and travel at the speed $c$.

These are the one-soliton solutions

\[u(x,t) = -\frac{c}{2}\operatorname{sech}^2

\left[\frac{\sqrt{c}}{2}(x-ct)\right],\]

one of which is plotted below.

The more particle-like qualities of solitons can be seen more clearly when there are multiple solitons, most simply illustrated with the two-soliton solution. At early and late times it looks like a superposition of one-solitons.

There are also non-soliton solutions, such as the periodic and difficult to pronounce cnoidal wave solution, which contains the Jacobi elliptic function $\operatorname{cn}$. This is like a ‘lattice’ of one-solitons placed next to each other.

For a wave to keep its shape, all the constituent sine waves must travel at the same velocity. In other words, there cannot be dispersion. This holds for many waves in nature: red light and blue light differ in wavelength, but travel at the same speed. The same is true of sound at different pitches. But it’s not so for typical surface waves, where waves with longer wavelength travel faster than those with shorter wavelength. Any lump of water built from many sine waves quickly falls apart.

What saves the solitary wave is that dispersion is delicately balanced by a nonlinear effect. This is neglected in rudimentary surface wave theory, which is a linearised theory. Linearity is what allowed us to think of the wave profile as being the sum of many independent sine waves, and the nonlinearity causes the sine waves to interact in potentially complicated ways.

Linearising is valid when the variations in height are small compared to the depth of the water. Our hands are small, and the lumps of water we can make aren’t big enough to bump us into the regime where the nonlinear effect becomes important. But boats are quite big, and can make swells of water large enough (relative to the depth of the canal) for nonlinearity to come into play.

The equation describing water waves which includes the nonlinear effect is known as the *KdV equation*, named after the Dutch duo Korteweg and De Vries. They weren’t the first to derive the equation but they found a solution that travelled with constant speed: the *one-soliton solution to KdV*.

Scott Russell’s solitary wave had been proven. He was sure the self-sustaining wave was hugely important, but it hadn’t caught the attention or imagination of many of his contemporaries. It wasn’t until the next century when the study of solitons really took off.

## Solitons

The next chapter of our soliton story picks up at the Los Alamos national laboratory in 1953. Four physicists (Fermi, Pasta, Ulam and Tsingou) were puzzling over what they were seeing: an animation simulating a vibrating string with nonlinear terms in its dynamics. The simulation was programmed on the Maniac computer (great name) by the most computer-savvy among them, Mary Tsingou.

They were interested in a hypothesis—called the ergodic hypothesis—that, roughly speaking, for systems which were mostly linear but perturbed by a nonlinear term, the initial energy would eventually (potentially after a long, long time) be evenly distributed over different degrees-of-freedom of the system. For their string, this would mean the amplitudes of all different frequencies would eventually be comparable.

The Maniac had also only recently been finished. With the Maniac at their door, the physicists tried what few could do before; study a problem computationally. They were looking in particular for a problem that would be easy to formulate, but which was intractable by hand or even mechanical computer.

In order to simulate the string on a computer, it had to be modelled by a discrete, finite number of points: in their case, 64 points. This is linear in the strain, and assuming that the strains are small, the next force to consider would be a quadratic one, and this was precisely the nonlinear dynamics considered, with neighbouring forces $F = k(\delta + \alpha\delta^2)$. They started the simulation with only an excitation in the fundamental frequency of the string. Early on, they saw the behaviour they expected, with successively higher frequencies of the string starting to receive small excitations. To their surprise, only one of the frequencies would have large excitations at any one time, starting from the fundamental frequency, then passing to the second mode, and then the third. This only continued among the first few modes, then the large excitation returned to the fundamental frequency, showing quasi-periodic behaviour.

This was unresolved until around twelve years later when two Americans, Kruskal and Zabusky, began to investigate the KdV equation computationally. They noticed that the continuum limit of the system studied by the Los Alamos four was described by the KdV equation.

By simulating the KdV equation using a more direct discretisation scheme, they found something amazing. No matter what initial configuration they prescribed for the water displacement height, the surface would break into a train of solitary waves with a profile like a one-soliton, each with their own amplitude and speed. Moreover, when these solitary waves collided, they would interact, maybe raising or dipping their peaks slightly as they passed one another, but then recover the shape and speed they had before the collision.

They knew they were onto something important, and they christened these resilient lumps *solitons*: ‘solit-’ for solitary, ‘-on’ to denote a particle, the suffix which appears at the end of a zoo of particle physics terms: electron, proton, hadron, baryon, and so on…

The particle-like behaviour of these self-sustaining lumps of energy is their ability to enter into, then emerge from, collisions with a well-defined shape and speed. The presence of both wave- and particle-like behaviour suggests quantum shenanigans, but there is nothing quantum here, just water waves.

In 1967 a team of four scientists, including Kruskal, found a construction of exact solutions to the KdV equation which consist of precisely $n$ solitons, which start at large separation, intersect and mingle, then regain their original size and shape, and drift apart.

The details of the construction of the explicit solutions are near magical. It turned out to borrow ideas from the scattering theory of quantum particles, then requires the application of a twisted functional transform to the scattering data. This transform is something like a nonlinear version of the Fourier transform, used extensively in applied maths due to its effectiveness in analysing signals.

The forward scattering problem in quantum mechanics is to determine the reflection and transmission of different frequency particles off a given potential. It’s like a mathematical description of how we see things: the object we look at is the potential, and the amount of light of different wavelengths that gets reflected into our eyes allows us to construct an image of that object.

In quantum mechanics, the potential is a function of space, while the reflection and transmission data is packaged into a function of frequency called the spectral data. The KdV equation has an associated scattering problem, and the complicated dynamics of KdV turns into a very simple time dependence for the spectral data.

On the KdV side, the wave profile $u$ becomes the potential in the scattering problem. In the scattering picture, we have a grasp on how the spectral data evolves, so to recover a solution $u(x,t)$ for KdV, we need to reconstruct the potential from its spectral data, which is an inverse scattering problem, and where the twisted transform comes in.

This brilliant but byzantine technique, known as the *inverse scattering method*, allowed the team to explicitly write down $n$-soliton solutions to KdV. Kruskal and Zabusky believed it was the presence of these solitons that meant the physicists’ string didn’t obey the ergodic hypothesis. The discovery of this inverse scattering method began a flurry of research into solitons.

## Beyond water waves

Shortly after the inverse scattering method was found for KdV, it was adapted to two other PDEs. Alongside the KdV equation, these PDEs have since taken on a celebrity status within the study of solitons.

Of the three, the *sine-Gordon equation* is the one which has been studied for the longest time. It began its life in the field of classical differential geometry. Surprisingly, solutions to the sine-Gordon equation correspond one-to-one with pseudospheres, which are surfaces of constant negative curvature immersed in three-dimensional space. Here, immersed means the surfaces might do funky things like intersect themselves or have sharp cusps. But that’s an article for another day.

The sine-Gordon equation has also drawn attention in several parts of physics, from particle physics and statistical physics to material science, where it was used to study screw dislocations in crystals. The one-soliton solution to sine-Gordon has the interesting property that it limits to $2\pi$ instead of zero as $x$ tends to infinity. That is, if we plot the sine-Gordon soliton as a function of position, it does not look like a lump, like the KdV soliton, but has an S shape.

## A PDE zoo

Soliton solutions are present for many PDEs. Here’s a brief *who’s who* of the most famous ones. Subscripts denote partial derivatives.

- The sine-Gordon equation:

\[ \varphi_{tt} – \varphi_{xx} + \sin\varphi = 0.\]

For this soliton, the limiting value at positive infinity in position is $2\pi$. - The nonlinear Schrödinger equation:

\[ \mathrm{i}\psi_t = -\frac{1}{2}\psi_{xx} + \kappa |\psi|^2 \psi.\] - Everyone knows that real canals are not one dimensional. A more realistic description of surface waves in a canal is the KP equation, a 2D generalisation of KdV:

\[

(u_t + uu_x + \varepsilon^2 u_{xxx})_x + \lambda u_{yy} = 0.

\]

As with the KdV equation there is a*lattice*of soliton solutions to the KP equation. This is what you can see at the Ile de Ré at the top of the article!

The sine-Gordon soliton is an example of a *topological soliton*. In fact, all solutions of the sine-Gordon equation have the property that the difference of limiting values between $x$ at positive and negative infinity is an integer multiple of $2\mathrm{\pi}$. This integer is called the winding number of the solution, and offers a glimpse of the interplay between topology and soliton theory.

The last of the three gold standard soliton PDEs is the *nonlinear Schrödinger equation*. The solitons of this PDE were the closest to having technological applications. Light signals in optical fibres are well modelled by solitons of this equation. Their ability to keep their shape was desirable for sending signals over vast distances, and solitons made it to lab trials before other technological advances in the field made them obsolete. In an alternate universe, the online edition of this magazine may have been brought to you courtesy of solitons.

There is a dizzyingly rich array of PDEs with soliton solutions. The majority of known PDEs admitting solitons are defined with only one spatial dimension, although of course more realistic systems incorporate two or three spatial dimensions. One such PDE is the *Kadomtsev–Petviashvili equation*, which describes a two-dimensional model of shallow water waves and generalises the KdV equation.

From humble beginnings in the Union canal, our soliton story has taken us worldwide, where they were one of the first phenomena explored using mathematical computing, and even into the materials and optics labs. The rich theory of solitons has sustained itself now for almost 200 years and solitons are now ubiquitous in mathematical physics, fulfilling Scott Russell’s vision. And there is still so much left to explore.