# Soliton

A soliton is a self-reinforcing solitary wave caused by nonlinear effects in the medium. Solitons are found in many physical phenomena, as they arise as the solutions of a widespread class of weakly nonlinear partial differential equations describing physical systems. The soliton phenomenon was first described by John Scott Russell (1808-1882) who observed a solitary wave in the Union Canal, reproduced the phenomenon in a wave tank, and named it the "Wave of Translation".

It is not easy to define precisely what a soliton is. Drazin and Johnson (1989) describe solitons as solutions of nonlinear differential equations which

1. represent waves of permanent form;
2. are localised, so that they decay or approach a constant at infinity;
3. can interact strongly with other solitons, but they emerge from the collision unchanged apart from a phase shift.

More formal definitions exist, but they require substantial mathematics. On the other hand, some scientists use the term soliton for phenomena that do not quite have these three properties (for instance, the 'light bullets' of nonlinear optics are often called solitons despite losing energy during interaction).

Some of the equations that describe solitons are the Korteweg-de Vries equation, the nonlinear Schrödinger equation and the sine-Gordon equation.

Some types of tidal bore, a wave phenomenon of a few rivers including the River Severn, are 'undular': a wavefront followed by a train of solitons. Other solitons occur as the undersea internal waves, initiated by seabed topography, that propagate on the oceanic thermocline. Atmospheric solitons also exist, such as the Morning Glory Cloud of the Gulf of Carpentaria, where pressure solitons travelling in a temperature inversion layer produce vast linear roll clouds.

In 1965 N.J. Zabusky of Bell Labs and M.D. Kruskal of Princeton University first demonstrated soliton behaviour in media subject to the Korteweg-de Vries equation (KdV equation) in a computational investigation using a Finite difference approach.

In 1967, Gardner, Greene, Kruskal and Miura discovered an 'inverse scattering transform' enabling analytical solution of the KdV equation. The work of Peter Lax has since extended this to solution of many related soliton-generating systems.

In 1973, Akira Hasegawa of AT&T Bell Labs was the first to suggest that solitons could exist in optical fibers. He also proposed the idea of a soliton-based transmission system to increase performance of optical telecommunications.

In 1988, Linn Mollenauer and his team transmitted soliton pulses over 4,000 kilometers using a phenomenon called the Raman effect, named for the Indian scientist Sir C. V. Raman who first described it in the 1920s, to provide optical gain in the fiber.

In 1991, a Bell Labs research team transmitted solitons error-free at 2.5 gigabits over more than 14,000 kilometers, using erbium optical fiber amplifiers (spliced-in segments of optical fiber containing the rare earth element erbium). Pump lasers, coupled to the optical amplifiers, activate the erbium, which energizes the light pulses.

In 1998, Thierry Georges and his team at France Télécom R&D Center, combining optical solitons of different wavelengths (wavelength division multiplexing), demonstrated a data transmission of 1 terabit per second (1,000,000,000,000 units of information per second).

In 2001, the practical use of solitons became a reality when Algety Telecom deployed submarine telecommunications equipment in Europe carrying real traffic using John Scott Russell's solitary wave.

## References

• N. J. Zabusky and M. D. Kruskal (1965). Interaction of 'Solitons' in a Collisionless Plasma and the Recurrence of Initial States. Phys Rev Lett 15, 240
• A. Hasegawa and F. Tappert (1973). Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion. Appl. Phys. Lett. Volume 23, Issue 3, pp. 142-144.
• P. G. Drazin and R. S. Johnson (1989). Solitons: an introduction. Cambridge University Press.

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