Early History: From Russell’s Discovery to KdV
This story began in 1834 when a 26-year-old Scottish engineer named John Scott Russell noticed a strangely behaving water wave while conducting an experiment to determine the most efficient design for canal boats. Let him tell us about this first encounter:
“I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped – not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, 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 [14 km/h], preserving its original figure some thirty feet long [9 m] and a foot to a foot and a half [30-45 cm] in height. Its height gradually diminished, and after a chase of one or two miles [2-3 km] I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation…” — J.S. Russell, Report on waves .
After his discovery, Russell studied these stable solitary waves in a more controlled environment by building a 9-meter long wave tank in his back garden and he made further important observations. One of his findings was that the higher the solitary wave is the faster it moves. Although Russell was convinced that his discovery is of great importance, not all of his contemporaries were on his side. For example, Airy was highly sceptical of solitary waves, because it seemed to contradict his shallow water theory, in which a finite amplitude wave cannot propagate without changing its profile. This tension was resolved by Boussinesq and Rayleigh, who showed (independently) that the nonlinear effect, which causes the change of shape in a travelling wave, can be balanced by dispersion thus solitary waves can form.
This is best captured by a nonlinear partial differential equation, which was (re)discovered by the Dutch mathematician Korteweg and his student de Vries in 1895 . The Korteweg-de Vries (KdV) equation reads
where is the unknown function in two variables, space () and time ().
Let us look for solutions which move from left to right with constant velocity and have a localized shape, i.e. we search for solutions satisfying
Plugging (2) into (1) gives us
which integrated with respect to leads to
Multiplying this by and integrating once more, turns it into
where and are constants of integration. The conditions of vanishing derivatives in (2) imply that leaving us with
Solving by separation of variables yields the solution
where . This is called a soliton and behaves as Russell’s wave of translation.
There are also 2-soliton solutions of (1), in which two solitary waves seem to be unaware of each other for large . In the middle (around ), they overlap and interact nonlinearly. Shortly after the interaction, they reappear with no apparent change in size or shape. Nevertheless, there is some mark of the interaction left on them, namely phase shifts.
Despite these early results, solitary waves fell into scientific obscurity for almost 70 years.
Digression: The Fermi-Pasta-Ulam Problem
When von Neumann‘s computer, named MANIAC-I, was completed in Los Alamos in 1951, Enrico Fermi, John Pasta and Stanisław Ulam promptly started to use it for numerical experiments on nonlinear physics and mathematics. As a test, Fermi suggested to study something to which he thought the answer was obvious: A fixed-end chain of 64 masses connected by springs exerting a nonlinear force between neighbouring weights. The force was taken to be not linear, but a quadratic or cubic function of the displacement. They started with a periodic vibration, a single sine wave, which, in case of a linear force, would oscillate in that mode indefinitely.
They expected that a nonlinear force, perturbing the periodic linear solution, will cause the oscillations to be of an ever-increasing complexity, i.e. get into states, where more and more Fourier modes are present. Physicists would say, that they expected the system to “thermalize”. They programmed the computer* and let it calculate. At first, they got the expected result and went out for lunch, but they had forgotten to turn the machine off! When they returned to the lab, they saw something incredible:
“…we indeed observe initially a gradual increase of energy in the higher modes as predicted. Mode 2 starts increasing first, followed by mode 3, and so on. Later on, however, this gradual sharing of energy among successive modes ceases. Instead, it is one or the other mode that predominates. For example, mode 2 decides, as it were, to increase rather rapidly at the cost of all other modes and becomes predominant. At one time, it has more energy than all others put together! The mode 3 undertakes this role. It is only the first few modes which exchange energy among themselves and they do this in a rather regular fashion. Finally, at a later time mode 1 comes back to within one percent of its initial value so that the system seems to be almost periodic.”
*It was Mary Tsingou, who has programmed the dynamics, ensured its accuracy and provided the graphs of the results.
For more on the FPU problem, see [3,4].
Revival: KdV as an Integrable Hamiltonian System
To make sense of the results of the FPU experiment Zabusky and Kruskal  considered the continuous version of it by shrinking the masses and springs to infinitesimal size, hence producing a line of deformable material. The corresponding partial differential equation can be transformed into the KdV equation. (For details, click here!) Thus Zabusky and Kruskal  started to study numerical solutions of the KdV-type equation
with and the periodic initial condition . At start, we have , so the dispersive term can be neglected, leaving us with the equation
Its solution can be given implicitly by
Such a tends to develop a discontinuity at at critical time . Yet, this is not what happens, since the term neglected at the beginning becomes significant, small wavelength oscillations form and the shock is eluded. Around these waves reach their final size and start to move as a train of solitons.
An amazing thing happens around . The spatially periodic solution of (8) through nonlinear interaction at arrive almost in the same phase and almost reconstruct the initial cosine curve. Hence is referred to as recurrence time. If this is the case in general, it explains the result of FPU.
Shortly after these initial results, Gardner et al.  gave a method of solving the KdV equation and formulated the following
Conjecture. Let be any solution of (1) which is defined for all and and which vanishes at . The there exist a discrete set of positive numbers — called the eigenspeeds of — and sets of phase shifts such that
It was proven by Lax  that for any pair of speeds there is a corresponding solution of KdV, which satisfies (11). The Conjecture was proven in many context and by many authors, including Ablowitz and Newell, Manakov, and Sabat.
Gardner, Kruskal and Miura  also made a remarkable discovery. The eigenvalues of the Schrödinger operator
are invariant in time if is a solution of the KdV equation (1). This means that there are infinitely many conserved quantities, i.e. functions of and its derivatives, which are constant along solutions . For example,
Indeed, Kruskal, Gardner and Miura constructed explicitly an infinite sequence of conserved quantities: . In 1968, Lax showed that the KdV equation is equivalent to an equation of operators (now called Lax equation) of the form
where is the Schrödinger operator (12) and is the skew-symmetric operator
(If two operators satisfy (14), they are called a Lax pair.) For this, solve the operator differential equation
and due to (14) and (16) we have
Therefore , i.e. and are similar, thus have the same eigenvalues. is isospectral.
where (13) and denotes the Fréchet derivative of the function :
The symplectic form is given by
and the Poisson bracket of two functions is defined by
It can be shown that for any two first integrals and one has
thus if both indices are odd or both of them are even, iterating this identity leads to
for some . If and has different parity, then we get
which by antisymmetry implies again that
Thus the conserved quantities are in involution, meaning that the KdV equation is completely integrable.
Closure: Other Soliton Equations and Applications
The ideas and results presented so far were all about the KdV equation. However, there are lots of other physically relevant nonlinear PDE’s, which have soliton solutions. Here we briefly describe two of them, namely the sine-Gordon and the nonlinear Schrödinger equation. We also mention several applications.
The sine-Gordon equation is a nonlinear PDE in dimensions, It has the form
(Its name is a pun due to Kruskal and refers to the low-amplitude () approximation, which is called the Klein-Gordon equation.) It can be interpreted as the equation describing the twisting of a continuous chain of needles attached to a flexible string. 1-soliton solutions are given by
where is arbitrary real, is real satisfying , and
Depending on the sign, we call (28) a kink () or an antikink ().
Similarly to KdV, 2-soliton solutions are as if 1-solitons would interact, namely we have (anti)kink-(anti)kink and kink-antikink collisions. But there is another type of 2-soliton solution, called breather, which looks like a coupled kink-antikink pair.
The sine-Gordon equation is relativistic, meaning that it is invariant under the Poincaré transformations of the -dimensional space-time. For details, see .
The nonlinear Schrödinger (NLS) equation reads
where is a complex-valued wave function and is constant. The equation is non-relativistic (Galilei invariant). This is also an exactly solvable Hamiltonian system . Its Hamiltonian is given by
Now let us list some applications of these soliton equations. The Korteweg-de Vries equation can be applied to describe shallow-water waves with weakly non-linear restoring forces and long internal waves in a density-stratified ocean. It is also useful in modelling ion acoustic waves in a plasma and acoustic waves on a crystal lattice.
The sine-Gordon solitons, kinks and breathers used as models of nonlinear excitations in complex systems in physics and even in cellular structures.
The nonlinear Schrödinger equation appears in the Manakov system, a model of wave propagation in fiber optics. The function represents a wave and NLS describes the propagation of the wave through a nonlinear medium. The second-order derivative represents the dispersion, while nonlinearity is in the term. The equation models many nonlinearity effects in a fiber, including but not limited to self-phase modulation, four-wave mixing, second harmonic generation, stimulated Raman scattering, etc.
- Russell, J.S., Report on waves, 14th meeting of the British Association for the Advancement of Science, 311-390, 1844.
- Korteweg, D.J., and de Vries, G., On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves, Philosophical Magazine 39(240): 422-443, 1895.
- Fermi, E., Pasta, J., Ulam, S. and Tsingou, M., Studies of nonlinear problems I, Los Alamos preprint LA-1940, 1955.
- Ford, J., The Fermi-Pasta-Ulam problem: paradox turns discovery, Phys. Rep. 213, 271-310, 1992.
- Zabusky, N.J., and Kruskal, M.D., Interaction of “solitons” in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 15, 240-243, 1965.
- Gardner, C.S., Greene, J.M., Kruskal, M.D., and Miura, R.M., Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett. 19, 1095-1097, 1967.
- Lax, P.D., Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Applied Math 21, 467-490, 1968.
- Zakharov, V.E. and Faddeev, L.D., Korteweg-de Vries equation: A completely integrable Hamiltonian system, Functional analysis and its applications 5, 280-287, 1971.
- Tao, T., An explicitly solvable nonlinear wave equation, blogpost, January 22, 2009.
- Zakharov, V.E., Manakov, S.V., On the complete integrability of a nonlinear Schrödinger equation, J. Theor. Math. Phys. 19, 551-559, 1974.