Making waves - singly
Background:'In the Hollow of a Wave off the Coast at Kanagawa', c. 1830, Katsushika Hokusai.
The discovery of solitons is one of the nicer stories of important science arising from an apparently insignificant observation. In August 1834, the naval engineer John Scott Russell was watching a horse-drawn barge on the Union Canal, Hermiston, Edinburgh, as part of his work on hull design. When the cable snapped and the barge suddenly stopped, Russel was impressed by what happened: 'A mass of water rolled forward with great velocity, assuming the form of 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. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some 30 feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel'.
Russell went on to a distinguished career, but despite his experimental work in a homebrew wave tank and a subsequent paper, his contemporaries never shared his view of the importance of what he called the 'Wave of Translation'. Partial vindication came later in the 19th century, when Boussinesq (1872) and Korteweg and de Vries (1895) showed how such self-reinforcing solitary waves arose from partial differential equations describing shallow water motion. But in 1965, Martin Kruskal and Norman Zabusky discovered a surprising result. An entirely different system of coupled harmonic oscillators, the Fermi-Pasta-Ulam experiment, also yielded the Korteweg-de Vries (KdV) equation.
Kruskal and Zabusky coined the term soliton for these travelling wave solutions, alluding to their particle-like properties. Whereas standard waves have peaks and troughs, and disperse as they travel, solitons consist of a single non-dispersing peak: linear effects spreading the waveform are exactly balanced by non-linear ones that focus it. Furthermore they show elastic interaction: two solitons of different sizes (and hence velocities) can pass through each other. This isn't, however, a normal wave collision where the heights add linearly; at the instant of superposition, solitons merge with a broader, lower peak.
While solitons were first recognised on the surface of water, the commonest ones in water actually happen underneath, as internal oceanic waves propagating on the pycnocline (the interface between density layers). Sailors have long known of bands of rough and smooth sea - 'tide rips' and 'slicks' - as well as the phenomenon of 'dead water', increased drag at the mouth of fjords. But post-1970s observation, particularly with satellite Synthetic Aperture Radar (SAR) and ship-borne Doppler Current Profiling, revealed these to be the surface manifestation of 'undular bores', subsurface packets of solitons. Typically travelling as 'waves of depression' - troughs only - of tens of metres amplitude and often kilometres in wavelength, they are initiated when tidal flow is perturbed by underwater features such as ridges. They're more than a nautical curiosity, as they affect acoustic propagation in the sea (of military interest); mix sediments and nutrients (of ecological interest); and put potentially dangerous stress on the legs of oil rigs.
Oceanic undular bores appear to arise by spontaneous breakdown of larger perturbations in systems governed by the KdV equation. This also occurs in the atmosphere, occasionally in the higher mesosphere, but most commonly in the lower atmosphere, the troposphere. A spectacular, much-publicised example is Morning Glory, an undular bore that forms seasonally over the Gulf of Carpentaria, Australia. In this case, the bore travels in an inversion layer - cold air trapped between the ground and warmer air above - each soliton generating a roll-shaped bank of cloud hundreds of kilometres long.
Bores, naturally, are better known as a river surface phenomenon. One spectacular example appeared around January 10th this year, when many newspapers worldwide published a photograph supposedly showing the instant of impact of the 2004 'Boxing Day Tsunami'. Other papers were suspicious. Lack of attribution played a part, as did the grins and umbrellas of the onlookers shown in companion photos that joined it on the e-mail circuit. Collective debunking, now summarised at the urban-myths website Snopes.com, soon revealed that the photos dated from 2002 and showed a tidal bore on the Qiantang River, Hangzhou, China. The largest in the world, with a wavefront up to 9 metres high, the Hangzhou bore (called the Black Dragon) is the subject of an annual tide-watching festival.
It's very often stated that tidal bores on rivers are the definitive example of solitons. The situation isn't so straightforward (and not helped by a mess of terminology from different fields). A bore is a general term for a moving step-discontinuity in water level, otherwise called a 'hydraulic jump'. The classic bore - regionally a mascaret, pororoca and aegir - arises in funnel-shaped estuaries that amplify incoming tides, the rapid rise propagating upstream against the flow of the river feeding the estuary. The profile depends on the Froude number, a dimensionless ratio of inertial and gravitational effects. At its most energetic, a bore has a turbulent breaking wavefront like an advancing waterfall; this is effectively a shockwave rather than a soliton. Slower bores take on an oscillatory profile with a leading wave (a dispersive shockwave) followed by a train of solitons.
An even more complex question is whether tsunami waves involve solitons. Tsunami waves generated by sharp localised impulses, such as meteorite strikes into the sea, generally do. Models of the late Jurassic Mjolnir impact by the Simula Research Laboratory, University of Oslo, predicted trains of solitary waves. A similar effect occurred with 1958 mega-tsunami at Lituya, Alaska, when rock dropped en masse into a bay following a landslide. An even smaller, but still dangerous, equivalent is the wash from high-speed super-ferries that produce soliton wakes.
Earthquake tsunamis such as the 2004 tsunami are initiated by broader scale impulses, and the spreading mechanism depends on the model. As long-wavelength water waves, tsunamis are generally modelled by the shallow water or long wave equations (a simplification of the Navier-Stokes equations for cases where the wavelength is much larger than the water depth). These incorporate Boussinesq and KdV equations as further approximations, both of which can give soliton solutions. Nevertheless, it's difficult to check models; tsunamis are near-impossible to observe in mid-ocean before they are modified by shore effects: the catastrophic increase in amplitude, and the steepening into bores. Sea level measurement shows, however, that tsunami waves, again unlike solitons in the strict sense, have both peaks and troughs. (A classic warning sign of an impending tsunami is the sea level dropping before the first wavefront arrives - a detail enshrined in the story of Hamaguchi Goryo, who in 1854 burned his rice harvest to warn villagers as the sea receded.)
Some analyses have suggested, however, that the regime may depend on travel distance: that tsunami waves close to the epicentre arrive as continuous sinusoidal waves, but long-distance ones may resolve into solutions of the KdV equation and travel as a train of solitary waves. In general, though, all I can conclude is that 'tsunami' and 'soliton' aren't terms widely associated in the scientific literature.
Soliton behaviour has also been seen in other fluid-like systems such as plasmas and flowing sand (barchan dunes have been observed to pass through each other). The Great Red Spot of Jupiter may also be some form of soliton. Following Kruskal and Zabusky's discovery of its broader applicability, soliton theory has extended well beyond the original application to fluids. Solitons appear in many other areas of physics governed by weakly nonlinear PDEs: for instance, the FitzHugh-Nagumo equations describing nerve impulse propagation; and the sine-Gordon equation in solid state physics and non-linear optics.
It's hard to predict where solitons will pop up next. One of their more intriguing manifestations is 'light bullets', spherical solitary waves (as predicted by the non-linear Schrödinger equation) in non-linear optical media excited by laser. On collision, they show various behaviours - they can split, fuse, alter path and tunnel through each other - that might be harnessed to make optical computers.
One optical effect that has reached fruition, however, is soliton-based communications. The idea was first suggested by Akira Hasegawa and Fred Tappert in 1973, when they showed theoretically that solitons could arise in optical fibres with a suitably tailored non-linear relation between light intensity and refractive index. Practical research took over a decade to catch up. Management of dispersion was one of the problems, and soliton technology, which sends laser data as 'pulse' vs. 'dark' states, has been in long-running competition for bandwidth and distance with the more traditional NRZ (non-return-to-zero) systems that send two intensity states with no zero state. Even so, a number of major telecoms providers, such as Marconi and Corvis Corporation, are now using soliton technology for ultra-long-haul (ULH) optical fibre networks that communicate over several thousand kilometres.
John Scott Russell, one feels, would have been delighted by the wealth of phenomena arising from his Wave of Translation. As Chris Eilbeck's Solitons Home Page at Heriot-Watt University says, 'It is fitting that a fibre-optic cable linking Edinburgh and Glasgow now runs beneath the very tow-path from which John Scott Russell made his initial observations, and along the aqueduct which now bears his name'.
Tackling the mathematics
Due to the analytical insolubility of the underlying partial differential equations, early work on solitons had to be done by numerical methods. This is still the mainstay of work with general starting conditions and geometries. Femlab, the finite element solver from Comsol, includes the Korteweg-de Vries as one of its standard equation-based models (images top and right, from Femlab 3.1). Using a time-dependent solver, it demonstrates a succession of faster solitons passing through a slower one, all reforming after the collision. The post-processed domain plot shows the solution extruded along the time axis.
Dr Magnus Olsson, product manager for Comsol's Electromagnetics Module, told me that Femlab 3.2 will stress time-dependent formulations in non-linear optics and electromagnetics, the type of media in which soliton effects are of increasing practical importance. For instance, in photonic crystals, which manipulate light internally through non-linear optical properties, the lossless transmission of a pulse round a right-angled bend in a waveguide can be modelled with sine-Gordon solitons. (This model, incidentally, is also applicable to the motion of dislocations in metals and the 'unzipping' of DNA).
The Mathematica image plots the exact solution for the interaction of two solitons, showing the characteristic phase shift and nonlinear superposition of amplitudes. It's now known that an isolated 2D soliton solution to the KdV equation has a sech^2 profile. This result arose from the analytical solution of the Korteweg-de Vries equation, found in 1967 using the inverse scattering transform of Gardner, Greene, Kruskal, and Miura.
An especial understanding came via the work of Hungarian-born mathematician Peter D Lax, who recently won the Abel Prize 2005 for his lifetime contributions to PDE solution. The transform method worked through several obscure steps (Professor Helge Holden, summing up Lax's work on the Abel Prize site, calls them 'miracles'). Lax reformulated the solution in terms of two operators, a Lax pair, that not only explained how the transform worked but also made a large family of other soliton-generating PDEs integrable.