What Is a Soliton? The Wave That Refuses to Spread Out

Drop a pebble in a pond and the ripples spread out and fade. Almost every wave does this. A soliton is the exception: a single, lone wave that travels with the same shape and speed, as if it were a solid object. Even stranger, two solitons can run into each other and come out the other side intact. That is the idea on our Soliton hoodie, whose slogan is “They Pass Through Unharmed”. This post branches from our explainer How Are Electricity and Magnetism Connected?, because some of the most useful solitons today are pulses of light.

Who discovered the soliton?

The Scottish engineer John Scott Russell. In his 1844 Report on Waves, reproduced by Heriot-Watt University, he described watching a boat being pulled quickly along a narrow canal by a pair of horses. When the boat suddenly stopped, the water it had set moving gathered at the bow and then rolled forward on its own, as he put it, “a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed” (Heriot-Watt University^[1]).

Russell followed it on horseback. He overtook it “still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height”, before losing it after a chase of one or two miles. He dated the encounter to August 1834 and called the phenomenon the “Wave of Translation”. Heriot-Watt notes it happened on the Union Canal at Hermiston, near Edinburgh, and that Russell later built a 30-foot wave tank in his back garden to study it (Heriot-Watt University^[1]).

Russell was convinced that his wave was fundamentally important, but nineteenth- and early twentieth-century scientists largely disagreed. The page notes that it was not until the mid-1960s, when scientists began to study nonlinear waves with digital computers, that his ideas were properly appreciated.

Why do ordinary waves spread out?

Two effects work against a lone hump of water. The first is dispersion: waves of different wavelengths travel at different speeds, so any hump, which is a mixture of many wavelengths, gradually smears out. The second is nonlinearity: in water, taller parts of a wave move a little faster than lower parts, so a hump tends to steepen at the front, like a wave about to break.

A soliton is what you get when these two effects exactly balance. Dispersion tries to flatten the hump, nonlinearity tries to sharpen it, and the result is a shape that does neither. Ordinary small waves simply add together when they overlap, a rule called superposition (OpenStax University Physics, §16.5^[2]). Solitons live in the nonlinear world where that rule fails, which is what makes their behaviour so unusual.

What is the KdV equation?

In 1895 the mathematicians Diederik Korteweg and Gustav de Vries published an equation for long waves in a shallow rectangular canal, and showed that it allowed a new type of long stationary wave (Korteweg & de Vries, 1895^[3]). In the scaled form mathematicians use today, the Korteweg–de Vries (KdV) equation is

u_t + 6u u_x + u_xxx = 0

Here u(x, t) is the wave height, x is position along the channel and t is time, all in scaled dimensionless units. The subscripts mean rates of change: u_t is how fast the height changes in time, u_x is the slope and u_xxx is the third derivative in space. The middle term, 6u u_x, is the nonlinear steepening; the last term, u_xxx, is the dispersion.

The equation has an exact solitary-wave solution:

u = (c/2) sech²[(√c / 2)(x − c t − x₀)]

where c is the wave’s speed (in the same scaled units), x₀ is its starting position and sech is the hyperbolic secant, a smooth single hump. The striking feature is that the height c/2 and the speed c are tied together. Our worked example: a soliton twice as tall travels twice as fast, and is also narrower, because its width scales as 1/√c.

Why are they called solitons?

The word first appears in the title of a 1965 paper by Norman Zabusky and Martin Kruskal, “Interaction of ‘Solitons’ in a Collisionless Plasma and the Recurrence of Initial States” (Zabusky & Kruskal, 1965^[4]). They solved the KdV equation numerically on a computer and watched solitary waves collide. APS Physics notes that the 1965 discovery was a direct result of the new computer technology available for simulations. The “-on” ending, like electron or proton, signals that these waves behave like particles.

In 1967 Clifford Gardner, John Greene, Kruskal and Robert Miura found a method for solving the KdV equation exactly. Their abstract says it can predict exactly the “solitons” that emerge from arbitrary starting conditions, and that solutions describing any finite number of solitons in interaction can be written in closed form (Gardner et al., 1967^[5]). In those exact solutions, two solitons that meet come out with their original shapes and speeds; the only lasting trace of the collision is a shift in their positions.

Where are solitons used today?

In optical fibres. Glass also has dispersion and a nonlinear response, so a light pulse can balance the two like a water soliton. In 1973 Akira Hasegawa and Frederick Tappert proposed that stable nonlinear optical pulses could travel in fibres with the right (anomalous) dispersion (Hasegawa & Tappert, 1973^[6]). In 1980 Linn Mollenauer, Roger Stolen and James Gordon sent 7-picosecond pulses through 700 metres of single-mode silica fibre at a wavelength of 1.55 µm and observed narrowing and splitting characteristic of solitons (Mollenauer, Stolen & Gordon, 1980^[7]). For more on light bending and slowing in materials, see Thermal Mirages Explained.

Shallow water is still the classic home of the solitary wave, which our Shoalon character celebrates. Heriot-Watt’s page lists solitons across science, from hydrodynamics to nonlinear optics and plasmas. More wave designs are in our Waves, Chaos & Matter collection.

Quick FAQ

What is a soliton in simple terms?

A single wave that keeps its shape and speed as it travels, because the spreading and steepening effects on it cancel out.

Who discovered solitons?

John Scott Russell observed one on the Union Canal in August 1834. The name “soliton” came from Zabusky and Kruskal in 1965.

Do solitons really pass through each other?

Yes, in the KdV equation they emerge with the same shapes and speeds, shifted slightly in position.

Are tsunamis solitons?

That is an open, case-by-case question. Some long waves in shallow water are well described by soliton-type equations, but a real tsunami is more complicated, so we do not call tsunamis solitons in general.

References

1. J. C. Eilbeck, “John Scott Russell and the solitary wave,” Heriot-Watt University (reproducing Russell’s 1844 Report on Waves). https://www.macs.hw.ac.uk/~chris/scott_russell.html
2. W. Moebs, S. J. Ling & J. Sanny, University Physics Volume 1 (OpenStax, Rice University), §16.5 “Interference of Waves”. https://openstax.org/books/university-physics-volume-1/pages/16-5-interference-of-waves
3. D. J. Korteweg & G. de Vries (1895), “On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves,” Philosophical Magazine (5th series) 39, 422–443 (DOI 10.1080/14786449508620739; metadata verified via Crossref). https://doi.org/10.1080/14786449508620739
4. N. J. Zabusky & M. D. Kruskal (1965), “Interaction of ‘Solitons’ in a Collisionless Plasma and the Recurrence of Initial States,” Phys. Rev. Lett. 15, 240–243 (DOI 10.1103/PhysRevLett.15.240; metadata verified via Crossref). https://doi.org/10.1103/PhysRevLett.15.240
5. C. S. Gardner, J. M. Greene, M. D. Kruskal & R. M. Miura (1967), “Method for Solving the Korteweg-deVries Equation,” Phys. Rev. Lett. 19, 1095–1097 (DOI 10.1103/PhysRevLett.19.1095; metadata verified via Crossref). https://doi.org/10.1103/PhysRevLett.19.1095
6. A. Hasegawa & F. Tappert (1973), “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (DOI 10.1063/1.1654836; metadata verified via Crossref). https://doi.org/10.1063/1.1654836
7. L. F. Mollenauer, R. H. Stolen & J. P. Gordon (1980), “Experimental Observation of Picosecond Pulse Narrowing and Solitons in Optical Fibers,” Phys. Rev. Lett. 45, 1095–1098 (DOI 10.1103/PhysRevLett.45.1095; metadata verified via Crossref). https://doi.org/10.1103/PhysRevLett.45.1095

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.