Chris Eilbeck was one of the more prolific producers of computer animated films at the Chilton site in the period 1973-1982. Many are concerned with his long-term interest in the study of solitons.
A soliton is a self-reinforcing solitary wave, that maintains its shape while it propagates at a constant velocity. John Scott Russell observed such a solitary wave in 1834 in the Union Canal in Scotland. Solitons are also found in non-linear crystals, plasma physics etc.
Solitons can interact with other solitons, and emerge from a collision unchanged or with a very small loss.
Although not the first of the Eilbeck films made, this film is a good introduction to several of the other films. It was made for showing at a Chilton Open Day, demonstrating how the computer is used in studying mathematical theories of waves of various sorts. It highlighted the FR80 microfilm recorder and the Edinburgh Dec10, a part of SERC's Interactive Computing Facility. It contains excerpts from some of the earlier films listed below.
In non-linear optics, in the propagation of highly intense pulses of coherent light through a dielectric medium, solitons are important because they travel through the medium without losing energy. The first method shown in the film is to consider the pulse as a slowly varying envelope modulating a resonant carrier wave. Theoretically, the evolution of this envelope is described by the Self-Induced Transparency (SIT) equations.
The second part looks at the the detailed structure of the pulse itself. The equations describing the structure are known as the Reduced Maxwell-Bloch (RMB) equations. Oscillating pulses similar to the 0-PI pulse solution shown before are called bions. It looks like a solitary wave envelope modulating a carrier wave but the carrier is no longer necessarily resonant as in the case of the SIT pulse. Bions behave like solitons and the film shows two bions colliding by letting a large fast pulse overtake a small slow one. The bions regain their original shape after collision The solutions of the RMB equations can easily be transformed into solutions of the sine-Gordon (SG) equation. The name bion never caught on - today this would be called a breather.
The Regularized Long Wave (RLW) equation (also called the BBM - Benjamin, Bona and Mahony equation) is a modified version of the Korteweg-de Vries (KdV) equation, and solutions at this resolution would look very similar, except for the square wave initial conditions.
Numerical studies of solitary wave solutions of the regularized long-wave (RLW) equation suggest that they exhibit true soliton behaviour, being stable on collision with other solitary waves. However later studies have shown a very small deviation from exact soliton behaviour.
The terms Boomeron and Zoomeron describe specific instances of solitons that have distinct features when they arise in various models. The Boomeron equation was introduced by Calogero and Degasperis. The film first shows a single boomeron with energy E > 0. The particle moves from right to left and then boomerangs back, thus the name. This is followed by the case where E < 0 and the particle moves right to left but in this case oscillates about a point X. The remainder of the film show the four cases of what happens if you have two boomerons which each have positive or negative energy and the point X for the two boomerons is the same or different.
Whereas boomerons are associated with the coupled Boomeron equation, zoomerons are associated with its descendant the scalar Zoomeron equation (ZE). The film shows the same set of cases as for the Boomeron film.
This film was made in collaboration with J D Gibbon and N C Freeman. See Freeman's paper: A Two-Dimensional Distributed Soliton Solution of the Korteweg-De Vries Equation in Proc Roy Soc Lond A366(1979) 185-204. The film shows:
The nervous system is made up of cells which communicate with each other by means of electrical signals. The conduction of electrical pulses in a nerve axon is an important problem in neurobiology. Under certain conditions, a train of pulses can travel along a semi-infinite nerve axon. If pulses travel along two parallel nerve fibres, does a pulse in one fibre induce a potential in the other fibre? Laboratory experiments suggest this happens.
A numerical study of the coupled nerve fibre problem verified and extended the perturbation theory of Luzader. Pulses on adjacent fibres can couple together with two possible stable pulse separations.
Pulse evolution on coupled nerve fibres, Eilbeck J C, Luzader S D, Scott A C, Bull Math Biol. 1981;43(4):389-400.
The film shows:
Film made in collaboration with Alwyn C Scott.
In 1973, Alexander Davydov constructed a quantum mechanical model for vibrational energy propagation down the alpha helix of a protein, showing how solitons could travel along the three spines (hydrogen-bonded chains) of the protein. A symmetric model is one where the energy is shared equally between the three spines. The strength of the coupling between vibration and lattice distortion depends on the nonlinear parameter X. Solitons can form when X > 0.45.
(see Davydov's soliton revisited, Scott, A C, Physica D, 51, 333-342, 1991).
The film shows:
The film shows a variety of analytic and numerical solutions of the perturbed and unperturbed sine-Gordon equations in one or two space dimensions.
The sine-Gordon equation as a model classical field theory. Caudrey P J, Eilbeck J C, Gibbon J D. Il Nuovo Cimento B, Vol. 25, No. 2, 02.1975, p. 497-512.
Internal dynamics of long Josephson junction oscillators. Christiansen P L, Lomdahl P S, Scott A C, Soerensen O H and Eilbeck J C. Applied Physics Letters Volume 39, 108-110 (1981).
Chaos in the inhomogeneously driven sine-Gordon equation. Eilbeck J C, Lomdahl P S, Newell A C. Physics Letters A, Volume 87, Issue 1-2, p. 1-4.
An important class of models relevant to a variety of applications is the Klein-Gordon type equations. One of the particularly interesting equations within this family is the so-called Φ4, featuring a wave equation with a cubic nonlinear (odd-power) polynomial added to it.
The film shows 2-kink collisions for different velocities V1 = -V2 = 0.9, 0.3, 0.24, 0.225, 0.21, 0.2 and 0.1.
It concludes by showing 3-kink collision where V1 = -V2 = 0.1 and V3 = 0.99.
Solitons and Nonlinear Wave Equations. Dodd R K, Eilbeck J C, Gibbon J D and Morris H C. Pp 640, London: Academic Press, 1982, Ch 10.
