Solitons

Historical information

First observation of soliton by J.S.Russel In 1834, John Scott Russell describes his wave of

translation. The discovery is described here in Russell's own words:

"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".

Explanation of Russel’s wave by Korteweg and de Vries Russell's experimental work seemed at odds with the Isaac Newton and

Daniel Bernoulli's theories of hydrodynamics. George Biddell Airy and George Gabriel Stokes had difficulty accepting Russell's experimental observations because they could not be explained by linear water wave theory. His contemporaries spent some time attempting to extend the theory but it would take until 1895 before Diederik Korteweg and Gustav de Vries provided the theoretical explanation

Resurrection of soliton

All that time the scientists was not interested in solitons. It was the only particular fact. At that time the physical world seems to be linear.

The resurrection of solitary wave was accidental. In 1952 Enrico Fermi asked D.Pasta and S.Ulam to conduct the numerical experiment: to calculate the vibration of the link of 64 weights connected by nonlinear spring

F=k(Δx)+b(Δx)2

They observe that the system after some time returns to the initial state. They made this discovery when by chance let computer MANIAC run for long time.

The KdV equation Latter the M.Kruskal and N.Zabuski had shown that such system is

described by KdV equation. ut=6 u ux-uxxx

This equation has set of solutions with unique properties, which now we call solitons.

The one-soliton solution

Three solitons solution Solitons can pass through each other without changing the form

(like particles)

Solitons nowadays

Today solitons take apart in many physicals areas, like hydrodynamics, quantum mechanics, particle physics and so on.

It was found a lots of equation with soliton-type solutions.

1962 British scientist Tony Skyrme suggested a very interesting idea, namely can one create a fermion from a scalar field? The answer is YES and the result is skyrmion.

A skyrmion is a homotopically non-trivial soliton solution of a nonlinear sigma model i.e., a particular case of a topological soliton.

If spacetime has the topology S3×R (for space and time respectively), then classical configurations are classified by an integral winding number because the third homotopy group: π3(SU(N)xSU(N)/SU(N))=Z

A crazy idea from Tony Skyrme

Baryon: Rotierendes Soliton

•Soliton: Kollektive Rotation um 8 Achsen im Spin-Flavour-Raum

•Wess-Zumino-Term schränkt Rotation ein 8, 10, 10, 27, ..

•Wess-Zumino-Term berücksichtigt axiale Anomalie der QCD

Pentaquark

M1890-180*Y] MeV

D. Diakonov, V. Petrov, and M. Polyakov, Z. Phys. A 359 (1997) 305.

Exotic: S=+1 Low mass: 1530 MeV Narrow width: ~ 15 MeV Jp=1/2+

The antiquark has a different flavor than the other 4 quarks.

A soliton wave.

A soliton wave is a confined energy phenomenon that travels at faster-than-light speeds, with potential applications in spacecraft propulsion. In the 24th century, Doctor Ja'Dar of Bilana III investigated soliton waves as a possible alternative to warp drive. The USS Enterprise-D attended the first field test of this technology in 2368. The test involved the generation of a soliton wave using 23 field coils on the surface of Bilana III, which would envelop a small, unmanned vehicle and push it into warp towards Lemma II. There, a sister facility would generate a scattering field to dissipate the wave and bring the test ship out of warp. Although the test began promisingly, displaying a 98% energy transfer efficiency of the soliton wave that was 450% more efficient than the Enterprise's own warp engines, the wave unexpectedly destabilized and manifested a subspace distortion that destroyed the test ship and damaged the Enterprise. Dr. Ja'Dar believed that a transient power imbalance was responsible. The Enterprise subsequently found that the wave was accelerating towards Lemma II, gaining energy in the process. It was estimated that by the time the wave arrived it would contain enough energy to destroy most of the planet. The wave had grown too powerful for the Lemma II facility to dissipate, but the Enterprise was able to disrupt the wave by detonating five photon torpedoes directly in front of it. (TNG: "New Ground")

A report from year 2368 http://memory-alpha.org/de/wiki/Die_Soliton-Welle