## Alex Kasman

The point of the "homemade soliton model" shown on my "An Introduction to Solitons" page is to explain the existence, but NOT the dynamics of solitons. In particular, after the discovery of "solitary non-linear waves" and before the modern understanding of solitons, it was argued that solitary waves would be very RARE. The argument was that their existence required a perfect balance of the distortion from the nonlinear terms and the dispersive terms in the equation, which would "obviously" hardly ever occur. In fact, as we now know, solitons are NOT rare, and the model is intended to show the way in which these two different effects on the wave can balance themselves automatically; i.e. there is a coupling between distortion and dispersion.

The model is very simple: take a long rod, and hang free swinging pedula from the rod at regular intervals. It is important that these pendula can swing around the rod freely, but do not move sideways. Put fixed weights at the end of the rods and connect them with rubber bands which are tight when the pendula all hang straight down. The precise weight and strength of the rubber bands is not important...and that is the point.

If the rotation of the rods is unaffected by friction (you can attempt to approximate this with lubricant) then the motion of the pendula is approximated by the discrete Sine-Gordon equation. The case in which all of the rods are hanging down (so there is not much tension on the rubberbands) is the zero solution. (If you have access to a Macintosh computer then I strongly recommend that you download the program "3D-filmstrip" by R. Palais at Brandeis University. It will show you an animation of an ideal model of this sort along with a description.)

To generate a single kink-soliton, start with the zero solution and take all of the pendula to the left of the center point and pull them over the top of the rod. You will get the one-soliton shape that is shown in the photograph because the rubberbands will pull the pendula near the center point together so that they stand up. Now we see the distortion and the dispersion! Gravity is pulling DOWN on the weights, attempting to make the soliton more narrow and "sharp", but the rubberbands are trying to pull the pendula together and "flatten" it out. The coupling is evident from the following observations: if you take the model from the first floor of a building up to the 20th floor, the strength of the gravitational pull on the model has changed. However, the solitary wave shape does not collapse! Similarly, you could have used stronger weights or weaker rubberbands, but everything will still work. Why? Because the more you pull on the rubberband the more it pulls back. So, it eventually finds an equilibrium point...and you see the solitary wave shape.

To generate a soliton/anti-soliton pair, start with the zero solution and pull JUST the center pendulum over the top of the rod until it is pointing straight down on the other side. You will have two "humps"...if your rod is long enough you can push these humps apart and they will stay where they are. But, if you let them come together they cancel each other and return to the vacuum solution. This is a good model for realizing the creation of an electron/positron pair since the continuous limit of this model, the Sine-Gordon equation, describes an electro-magnetic field under proper assumptions with the solitons playing the role of the particles through "bosonization".

I hope that this description is adequate for your interests. If I have not been clear or if you have any further questions, please let me know.

Contact Information:
Department of Mathematics
College of Charleston
66 George Street
Charleston, SC 29424-0001