The solitons a hump already contains

The pulse two failures keep alive is a solitary wave in which dispersion, spreading the pulse, and nonlinearity, steepening it, cancel exactly. The cancellation is precise: for each width there is one height, and the pulse of that shape travels unchanged, at a speed proportional to its height.

That leaves the obvious question. A wave in a canal, a pulse in an optical fibre or a disturbance in the ocean is almost never launched with the exact shape that balances. What does a hump of the wrong height for its width do? It might keep a distorted shape, or spread away and vanish, or break. It does none of those. It separates into a definite number of solitons of definite heights, which run apart because taller solitons are faster, and it sheds whatever is left over as a ripple that disperses. The number and the heights are fixed before the hump moves at all — and the way to read them off is to solve a problem from a completely different part of physics.

A hump coming apart #

The equation is the Korteweg–de Vries equation in its standard form, $u_t + 6uu_x + u_{xxx} = 0$, whose solitons are $u = 2\kappa^2\,\mathrm{sech}^2(\kappa(x - 4\kappa^2 t))$: height $2\kappa^2$, speed $4\kappa^2$, width $1/\kappa$. Start it instead from a hump of height 6 and unit width, $6\,\mathrm{sech}^2 x$, which is three times too tall for its width to be a soliton.

The integration is pseudo-spectral and conserves the hump’s area to fifteen decimal places. By the time 0.15 the hump has already leaned forward and grown a shoulder behind it. By 0.35 the shoulder is a separate crest. By 0.6 there are two crests, of heights 8.00 and 2.00, the taller one travelling four times as fast, and each has the exact shape of a soliton of its height. Nothing about the starting hump suggested either number. Its height was 6; neither crest is 6, and the taller one is taller than anything present at the start.

The hump read as a well #

In 1967 Clifford Gardner, John Greene, Martin Kruskal and Robert Miura found where the numbers come from, and the answer was a surprise. Take the starting shape $u_0(x)$, turn it upside down, and treat it as the potential in the time-independent Schrödinger equation, $-\psi'' - u_0(x)\,\psi = E\,\psi$ — the equation for a quantum particle trapped in a well. A hump becomes a well. A well holds bound states at negative energies, $E = -\kappa^2$, only at particular values.

The levels were found without any formula, by integrating trial solutions across the well and counting how many times each one crosses zero — a count that jumps by one each time the trial energy passes a bound level. The hump of height 6 holds two levels, at depths 4 and 1. Twice those depths are 8 and 2, the heights of the two crests in the first figure. Each bound level of the well becomes one soliton, of height $2\kappa^2$, and the number of solitons a hump releases is the number of levels it holds.

The reason is one of the deepest facts in the theory of nonlinear waves. As $u$ evolves under the Korteweg–de Vries equation, the Schrödinger problem built from it changes shape at every moment — and its bound levels do not move. The evolution is isospectral: the levels are constants of the motion. At the start they belong to one smooth hump; at late times, when the solitons have separated far enough not to overlap, the well has become a set of separate soliton-shaped wells, each of which holds exactly one level. Matching the two ends gives the heights. That constancy is also the source of the equation’s infinite family of conserved quantities, which is what makes two solitons pass through each other and emerge intact.

Why a linear problem can govern a nonlinear one #

It is worth being precise about how odd this is. The Korteweg–de Vries equation is nonlinear: two solutions do not add to make a third, a tall hump behaves differently from two short ones, and nothing like a superposition principle holds. The Schrödinger equation used to read the hump is linear in its wavefunction, and its levels are found by the methods used for any well — the same counting that fixes the energies a box allows and the levels that split as wells approach.

The connection is a structure Peter Lax identified in 1968. The Schrödinger operator built from $u$ can be paired with a second operator such that the rate at which the first changes in time equals the commutator of the two — and writing that condition out term by term reproduces the Korteweg–de Vries equation exactly. An operator that evolves by a commutator keeps its eigenvalues, the way a matrix rotated in time keeps its eigenvalues while its entries change. So the nonlinear wave equation is, in disguise, the statement that a linear operator is being rotated without being stretched.

That is the same kind of fact as a symmetry handing over a conservation law, with a far larger symmetry behind it. Each level is a conserved quantity, and so is every other function of the operator’s spectrum, which is why the equation has infinitely many conservation laws where an ordinary nonlinear equation has two or three. It is also why the equation is special: nearly every other nonlinear wave equation has no such pair, and for them no linear problem reads off the outcome.

The heights of the form $N(N+1)$ — 2, 6 and 12 in the figure — are special. They hold exactly $N$ levels at depths $1, 4, 9, \ldots$ and they are reflectionless: a wave sent across such a well passes straight through without reflecting, however low its energy. Those humps shed no ripple at all, and resolve completely into solitons. A hump of height 4 holds two levels at depths 2.44 and 0.32 and reflects some of every wave, and the part of the hump that corresponds to reflection leaves as a dispersive tail.

How many solitons #

The count depends on the hump’s size relative to its width. For humps shaped as $\mathrm{sech}^2 x$ there is a closed form, and for any other shape the levels can simply be counted.

For the $\mathrm{sech}^2$ shape a new level appears each time the height passes $N(N+1)$, and the formula was checked against node counting at eight heights. Gaussian humps, with no special relation to the equation, hold one level at height 1, two by height 3, three by height 11, four by 20. The steps come at different heights for a different shape, and the staircase is the same kind of staircase.

The left end of the staircase says something that is easy to pass over. Every positive hump, however low, holds at least one level. That is a theorem about wells in one dimension: any attractive well, however shallow, binds at least one state — unlike a well in three dimensions, which must reach a threshold depth before it holds anything — with a binding energy that goes to zero as the square of the well’s area. So in the Korteweg–de Vries equation no positive disturbance disperses away entirely. The smallest bump leaves a soliton, of a height proportional to the square of its area, and the rest of it spreads.

For a low hump the soliton it leaves can be computed without solving anything. A shallow well binds its single level at $\kappa \approx \tfrac12\int u_0\,\mathrm{d}x$, so the soliton’s height is about half the square of the hump’s area. A hump of height 0.2 and unit width has an area of 0.4 and leaves a soliton of height 0.08 — less than half as tall, five times as wide, and travelling at two fifths of the speed a soliton of the hump’s original height would have. The rest of the hump’s material spreads as ripple. So a weak disturbance is mostly radiation with a small soliton inside it, and a strong one is mostly solitons with a small ripple left over, and the crossover is where the hump’s height times its width squared is of order one — the same comparison of nonlinearity with dispersion that sets a soliton’s shape in the first place.

The comparison also explains why solitons are not seen from every disturbance in a real channel. A disturbance must travel far enough for the solitons to separate from one another and from the ripple, and the distance needed grows as the disturbance weakens. How far a dispersive packet travels before it spreads and how far a steep front travels before it breaks are the two lengths that decide which happens first.

A trough releases nothing #

The same reasoning makes a prediction about disturbances of the other sign, and it is stark.

A trough, turned upside down, is a barrier rather than a well, and a barrier holds no bound states. So a trough releases no solitons at all. The integration bears that out: by time 0.6 its deepest point has risen from −4 to −1.46, and it has become a lengthening train of ripples running backwards, with no crest in it that keeps its shape. For this equation, a depression of any depth is entirely radiation.

That asymmetry between elevations and depressions is not a peculiarity of mathematics. For water waves in a shallow channel the Korteweg–de Vries equation has the sign it has here, and solitary waves of elevation are what John Scott Russell followed on horseback in 1834; solitary waves of depression do not occur. In a fluid layered with lighter water over heavier the effective sign can reverse, and then depressions form solitons and elevations disperse — which is why internal waves in a stratified sea are often observed as trains of troughs.

Heights nobody designed #

The strongest test is a hump with no relation to the equation, whose soliton heights could not be guessed.

A Gaussian of height 8 holds two levels, at $\kappa = 2.356$ and $1.252$, predicting solitons of heights 11.10 and 3.14. Integrating the full nonlinear equation to time 0.6 leaves crests of 11.10 and 3.14. The linear problem, solved once at the start, has predicted the outcome of a nonlinear evolution.

The same numbers settle where the hump’s material goes, and the answer is not what the picture suggests. The equation conserves the area under $u$ exactly, and a soliton of parameter $\kappa$ has area $4\kappa$. The Gaussian hump has area $8\sqrt{\pi} = 14.18$; its two solitons have areas $4 \times 2.356$ and $4 \times 1.252$, which add to 14.43. The solitons carry away more than all of the hump’s area, and the ripple left behind must have a net area of −0.25: it is, on balance, a depression. The hump of height 6 is the tidy case — area 12, solitons of area 8 and 4, ripple of area zero, as a reflectionless well requires — and the hump of height 4, with solitons of total area 8.49 from a hump of area 8, sheds a net depression of −0.49. Positive disturbances pay for their solitons by leaving the water behind them slightly lower than it started, which is also what a trough does on its own.

John Scott Russell saw the phenomenon before anyone could explain it. In the tank experiments he made after following his first solitary wave along a canal, he released heaps of water of different sizes at one end of a long channel, and he recorded that a heap too large to form a single wave did not form one larger wave: it separated into a leading wave and smaller ones behind it, each travelling on at its own speed. The speed he measured for a single wave, rising with its height above the undisturbed depth, is the speed that depends on the length corrected for amplitude, and the separation he saw is the figure above in water.

The pattern that results in nature has a name. A disturbance that releases several solitons sends them out rank-ordered: tallest first, because tallest is fastest, followed by progressively smaller ones, with the ripple trailing. Satellite images of the ocean show exactly that in the internal waves that radiate from tidal flow over sills and ridges — packets of solitary waves with the largest at the front and the spacing between them widening with distance. A pulse in an optical fibre too intense for its duration does the corresponding thing under the fibre’s own soliton equation, splitting into several solitons, and that fission is part of how a short intense pulse broadens into a spectrum far wider than it started with.

What makes the trick work, and where it stops #

The equation is integrable, and almost no equation is. The constancy of the levels is a special property of the Korteweg–de Vries equation and a handful of others, including the nonlinear Schrödinger equation of optical fibres and the sine-Gordon equation. Add a small damping, a variable depth or a second horizontal dimension and the levels drift, the soliton count is no longer exactly conserved, and the prediction becomes approximate. The chain of oscillators in the energy that refuses to be shared is close to integrable, which is why it behaves so strangely, and not exactly integrable, which is why it eventually does share its energy.

The water-wave version is itself an approximation. The Korteweg–de Vries equation describes waves that are long compared with the depth, small compared with the depth, and travelling in one direction. A real wave in a canal satisfies all three only roughly, and a tsunami reaching a shelf violates the second.

The ripple is not only noise. The continuous part of the Schrödinger problem — how the well reflects waves at positive energy — determines the dispersive tail exactly, and the full inverse scattering transform reconstructs the whole solution at any time from the levels and that reflection. The figures use only the levels, which is the part that survives.

The shift in position the snapshots hide #

The snapshots show where each crest has got to and not what happened to its position. Solitons that separate out of a single hump are shifted in position relative to where a lone soliton of the same height would be, by an amount set by the other levels, and the same shifts appear when two solitons collide. Those phase shifts are real, measurable and entirely invisible in a figure without a reference soliton to compare against.

Nor can the well figures show why a nonlinear problem should be governed by a linear one. The statement that the levels do not move is a theorem about a pair of operators whose commutator reproduces the equation, and it can be verified — the figures verify its consequence to a tenth of a per cent — without being made to look inevitable.

Still open: what a sea of solitons does #

A single hump releases a handful of solitons. A random, extended wave field in an integrable system contains an enormous number of them, interacting continually, and its statistics — how often very large crests occur, how the energy is shared between solitons and ripple, what a steady state looks like — are the subject of what is called soliton gas or integrable turbulence. The theory of such gases, built on the same levels, has made predictions that have been compared with laboratory experiments in water tanks and optical fibres since the late 2010s, and the comparison is promising and incomplete. Whether the extreme waves occasionally observed at sea owe anything to the nearly integrable dynamics of deep-water waves, rather than to linear focusing of many small waves, is a question on which the two descriptions give different answers and the data do not yet choose between them.

The habit worth carrying away is to look for what does not change. When a nonlinear evolution has a linear problem whose spectrum it leaves fixed, the outcome is written in that spectrum from the start — and the right question about an unfamiliar disturbance is not how it will move but which quantities its motion cannot alter.