Interaction Between Two Solitons

# Acoustics and Vibration Animations

Daniel A. Russell, Graduate Program in Acoustics, The Pennsylvania State University

All text and images on this page are ©1996 by Daniel A. Russell and may not used in other web pages or reports without permission.

The content of this page was originally posted on October 24, 2009. The HTML code was modified to be HTML5 compliant on January 27, 2014.

# Interaction Between Two Solitons

The animations on this page were inspired after reading: Roger Knobel Introduction to the Mathematical Theory of Wave Motion, (American Mathematical Society, 2000), pp. 31-35.

## Solitons (Solitary Waves)

In 1834, John Scott Russell, a Scottish naval enginer, was observing the passage of a boat along a canal and noticed a very strange type of wave traveling along the canal. His famous (and oft repeated) summary of the event[1] states . . .

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, preserving its original figure some thirty 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. 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.

Fascinated by what he had observed, Russell constructed a 30-foot wave tank in his back yard and carried out experiments. He made the following observations:

• These "solitary" waves are stable, and can travel over very large distances without changing their shape, neither decreasing in amplitude nor breaking as waves in water often do.
• The speed of the wave depends on the height of the wave.
• These waves don't obey superposition. When a taller (faster) wave overtakes a shorter (slower) wave, they don't combine and add together. Instead they appear to swap places with the faster wave appearing to jump through the slower one.

The mathematical theory explaining the existince of these solitary heaps of water was first deveoped by Boussinesq[2] in 1871 and Rayleigh[3] in 1876. However, it is the form of the equation and its solutions as developed by Koreweg and de Vries[4] in 1895 that is most commonly used to describe the waves that have become known as "solitons". The equation of motion for these waves can be written in dimensionless form as $$\frac{\partial x}{\partial t} + u\frac{\partial u}{\partial x} + \frac{\partial^3 u}{\partial t^3}=0$$ with wave functions satisfying this equation of motion having the form $$u(x,t) = 3c\,\hbox{sech}^2\biggl(\frac{\sqrt{c}}{2}(x - ct)\biggr)$$ where the wave speed c depends on the height of the wave pulse.

## Two solitons traveling in the same direction

The animation at left shows two solitons travelling in the same medium and in the same direction. Each soliton looks like a wave pulse that maintains a constant shape as it travels. The surface of water is a dispersive medium, which means that the wave speed is not constant and a wave pulse will change shape as it travels. Water surface waves travel with a speed that depends on height so that the crest of the wave travels faster than the trough and the wave will eventually break and fall over. In addition, the wave wil usually lose some energy as it travels through natural dissipative processes and the amplitude will decrease. However, solitary waves obey a nonlinear wave process in which the nonlinear effects offset the dissipative and dispersive effects and the wave propagates with constant shape and speed.

The speed of these solitary waves depends on the height of the wave, so the taller wave is faster than the shorter wave. Thus, the taller wave overtakes and passes the smaller wave.

## Collision between two solitons traveling in the same direction

Normal, linear waves obey the principle of superposition - which means that the amplitudes of two waves traveling through the same medium at the same time simply add together. Linear waves interfere with each other by simply adding their amplitudes together. Solitary waves, however, do not obey the principle of superposition, and instead of interacting through interference and simple addition, they collide in a nonlinear and complicated manner. The double soliton solution[5] is not the simple sum of the two individual solitons.

The animation at left shows what happens when these two solitons meet in the same medium. The top trace (thin gray line) shows what we would expect to see if the wave pulses simply interfered with each other through superposition. Of course, if these were linear waves, they would both be traveling with the same speed the tall one would not overtake the shorter one. The bottom trace (thick black line) shows what actually happens to these two solitary waves. As the faster, taller pulse catches up to the shorter, slower pulse, the two do not simply add together. Instead, the taller pulse appears to jump through the shorter one and they switch places. After they "collide" they keep moving each with their own speeds but they are not in the relative locations we would have expected the to be in had they simply passed through eachother. The contour plot at right shows the traces of the two pulses as they collide, and it is easy to see that the paths of the two pulses appear to jump and change places rather than just pass through each other.

### References

1. John Scott Russell, Report on Waves, Report of the 14th Meeting of the British Association for the Advancement of Science, (1844), pp.311-390.
2. M.J. Boussinesq, "Theorie de l'intumescence liquid appellee onde solitaire ou de translation, se propageant dans un canal rectangulaire." Comptes Rendus Acad. Sci., 72 755-759 (1871).
3. J.W.S. Lord Rayleigh, "On Waves," Phil. Mag., 1, 257-279 (1876).
4. 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 (1895)
5. G.B. Whitham, Linear and Nonlinear Waves (John Wiley & Sons, 1976), pp. 580-583.