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## Tag Archives: solitons

As kids tend to discover rather early on, it’s fun to throw stones into water. As the stone hits the water, it gives it energy and creates ripples that move outwards in concentric circles. These ripples carry the energy outwards, and as they move further out they eventually die out and the water is smooth again (this is true even if you ignore the effect of friction). The behavior of the water is well described by the wave equation — very likely the first field equation that people are taught about in physics. In one dimension, it describes waves on a string or electric and magnetic waves. In two dimensions, it describes water waves. In three, sound waves. Waves are ubiquitous in our daily lives.

In contrast, when you kick a soccer ball, the energy you give it doesn’t spread out. It stays with the ball as it rolls away. The difference between soccer balls and water waves is that in the former case the energy density is localized, and in the latter it dissipates. So the question arises: are there any wave type phenomena with a localized, non-dissipative energy density?

Solutions of field equations that have this property are called solitary waves. We do hear of such objects from time to time – moving lumps of water that don’t die out like waves do – and usually with catastrophic consequences. These are the tsunamis. The physics of tsunamis is rather involved, but certain kinds of tsunamis involve solitary waves – those that are caused due to sharp localized impulses such as an asteroid collision.

If these solutions have the further property that they retain their shapes after collision with each other, they are known as solitons. These objects show up in theories where non-linear interactions balance out the tendency to disperse, and they give rise to lots of interesting physics!

The discovery of the solitary wave is the kind of story that physicists love to tell. It’s a example of Isaac Asimov’s principle –  “The most exciting phrase to hear in science, the one that heralds new discoveries is not ‘Eureka!’ but ‘That’s funny…'”. The story is described in this article on the history and application of solitons:

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’.

I find it extremely cool that an idea that started off with a seemingly mundane observation of horse drawn boats can predict the existence of magnetic monopoles in certain gauge theories, or can explain flux quantization in superconductors! The important point is that rather than write off his observation as a quirk of nature, John Russell was the first person to follow through and experiment to try to understand this phenomenon.

Here’s a supremely ambitious vague plan of study for this semester. I want to spend the next 4 months thinking about some (subset) of the following things:

1. The path integral approach, and it’s relation to topics like solitons, instantons, monopoles. Applications to high energy and condensed matter.

2. Conformal field theory. Dualities. What is the ADS / CFT correspondence and the holographic principle?

3. Renormalization and the Renormalization group. Applications to high energy and condensed matter. Wilson loops. Renormalization and ricci flow.

The plan is subject to plenty of change as I piece together what’s going on. I’ll add useful references as I find them.

Here’s a collection of literature on these subjects: