# Tag Archives: Solitons

## Origin of the solitary wave

Back in 1834, a Scottish engineer named John Scott Russell noticed a strange kind of wave that propagated away from a boat that had made a sudden stop. He describes how he chased the wave on horseback along the river for about two miles! Here is his account from this manuscript:

“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”.

What he called a “Wave of Translation” is now known as a solitary wave or a soliton. Here is an excellent Youtube video of a solitary wave that has been generated in a lab setting. Remarkably, in the video, when two solitary waves meet, they appear to pass right through one another.

The strange thing about the solitary wave, is that it can propagate for miles at a time without breaking or disappearing, i.e. it is much more stable than the garden-variety (ocean-variety?) wave.

There are two concepts that are necessary to go over in order to understand the solitary wave. These are (i) non-linearity and (ii) dispersion.

Dispersion occurs when the phase velocity and the group velocity of a wave differ. If one forms a Gaussian wavepacket from a dispersive medium, then one would expect the amplitude to decrease and for the wave to broaden over time. You can see this effect occurring for the waves in the boat’s wake in this Youtube video. To those more familiar with quantum mechanics than fluid mechanics (which is usually the case for most physics students!), the same thing happens for a Gaussian wavepacket for an electron wavefunction, but does not occur for a wavepacket of light. It is important to note that this broadening and disappearing can happen in a medium that is not viscous. Viscosity has nothing to do with the dispersion of the water wave nor of the electron.

Non-linearity in water waves, on the other hand, is essentially what causes waves to “break”. The water at larger heights moves at a faster speed than the water at lower heights which results in a multi-valued function. Here is an instructive image which depicts well what I’m trying to get across (taken from here):

Also, here is Youtube video of breaking ocean waves to just soothe your soul.

So why is the solitary wave so stable and what is preventing it from breaking or disappearing? It turns out that the dispersion and non-linearity amazingly cancel each other out in order to sustain the solitary wave and allow it to propagate much further than regular waves.

Feynman said in his Lectures on Physics:

[water waves] that are easily seen by everyone and which are usually used as an example of waves in elementary courses […] are the worst possible example […]; they have all the complications that waves can have.

While this is true, it is these complications that give rise to some startling phenomena, including the solitary wave.

## An Integral from the SSH Model

A while ago, I was solving the Su-Schrieffer-Heeger (SSH) model for polyacetylene and came across an integral which I immediately thought was pretty cool. Here is the integral along with the answer:

$\int_{0}^{2\pi} \frac{\delta(1+\mathrm{tan}^2(x))}{1+\delta^2\mathrm{tan}^2(x)}\frac{dx}{2\pi} = \mathrm{sgn}(\delta)$

Just looking at the integral, it is difficult to see why no matter what the value of $\delta$, the integral will always give +1 or -1, which only depends on the sign of $\delta$. This means that if $\delta=1,000,000$ or if $\delta=0.00001$, you would get the same result, in this case +1, as the answer to the integral! I’ll leave it to you to figure out why this is the case. (Hint: you can use contour integration, but you don’t have to.)

It turns out that the result actually has some interesting topological implications for the SSH model, as there are fractional statistics associated with the domain wall solitons. I guess it’s not so surprising that an integral that possesses topological properties would show up in a physical system with topological characteristics! But I thought the integral was pretty amusing anyhow, so I thought I’d share it.

Aside: For those who are interested in how I arrived at this integral in the SSH model, here are some of my notes. (Sorry if there are any errors and please let me know!) Also, the idea of solitons in the SSH model actually bears a strong qualitative resemblance to the excellent zipper analogy that Brian Skinner used on his blog.