Soliton

In mathematics and physics, a soliton is a self-reinforcing solitary wave (a wave packet or pulse) that maintains its shape while it travels at constant speed; solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. ("Dispersive effects" refer to dispersion relations, relationships between the frequency and the speed of waves in the medium.) Solitons are found in many physical phenomena, as they arise as the solutions of a widespread class of weakly nonlinear dispersive 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 (a canal in Scotland), reproduced the phenomenon in a wave tank, and named it the "Wave of Translation".

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Definition

A single definition of a soliton is difficult to procure. Drazin and Johnson (1989) ascribe 3 properties to solitons:

1. They are of permanent form;
2. They are localised within a region;
3. They can interact with other solitons, and emerge from the collision unchanged, except for 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).

Explanation

To see how dispersion and non-linearity can interact to produce permanent and localized wave forms, consider a pulse of light traveling in glass. This pulse can be thought of as consisting of light of several different frequencies; since glass shows dispersion, these different frequencies will travel at different speeds and the shape of the pulse will therefore change over time. However, there is also the non-linear Kerr effect: the speed of light of a given frequency depends on the light's amplitude or strength. If the pulse has just the right shape, the Kerr effect will exactly cancel the effect of dispersion, and the pulse's shape won't change over time: a soliton. See soliton (optics) for a much more detailed description.

Many exactly solvable models have soliton solutions, including the Korteweg-de Vries equation, the nonlinear Schrödinger equation, the coupled nonlinear Schrödinger equation, and the sine-Gordon equation. The soliton solutions are typically obtained by means of the inverse scattering transform and owe their stability to the integrability of the field equations. The mathematical theory of these equations is a broad and very active field of mathematical research.

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 pycnocline. 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. The recent and not widely accepted soliton model in neuroscience proposes to explain the signal conduction within neurons as pressure solitons.

A topological soliton, or topological defect, is any solution of a set of partial differential equations that is stable against decay to the "trivial solution" due to topological constraints, rather than due to the integrability of the field equations. The constraint arises almost always because the differential equations must obey a set of boundary conditions, and the boundary has a non-trivial homotopy group, preserved by the differential equations. Thus, the solutions of the differential equations can be classified into homotopy classes. There is no continuous transformation that will map a solution in one homotopy class to another; thus the solutions are truly distinct, and maintain their integrity, even in the face of extremely powerful forces. Examples of topological solitons include the screw dislocation in a crystalline lattice, the Dirac string and the magnetic monopole in electromagnetism, the Skyrmion and the Wess-Zumino-Witten model in quantum field theory, and cosmic strings and domain walls in cosmology.

History

In 1834, John Scott Russell describes his wave of translation.

In 1965 Norman Zabusky of Bell Labs and Martin 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 on Lax pairs and the Lax equation has since extended this to solution of many related soliton-generating systems.

Solitons in fiber optics

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

Solitons in a fiber optic system are described by the Manakov equations.

In 1987, P. Emplit, J.P. Hamaide, F. Reynaud, C. Froehly and A. Barthelemy, from the Universities of Brussels and Limoges, made the first experimental observation of the propagation of a dark soliton, in an optical fiber.

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 per second 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. (The founders of Algety Telecom were a team of engineers in France Telecom's R&D Center (CNET) who had been working for nearly 10 years to develop soliton technology.In 2000 Corvis Corp signed an agreement to acquire Algety Telecom, in an all-share transaction, but later closed the Algety subsidiary.Corvis was then purchased by Broadwing, and Broadwing subsequently were purchased by Level3) The present status of using optical soliton for communication is not clear and more information is needed.)^{[citation needed]}

For some reasons, it is possible to observe both positive and negative solitons in optic fibre. However, usually only positive solitons are observed for water wave.

Bions

The bound state of two solitons is known as a bion.

See also

• Clapotis
• Freak waves may be a related phenomenon.
• Oscillons
• Q-Ball a non-topological soliton
• Soliton (topological).
• Soliton (optics)
• Soliton model of nerve impulse propagation
• Spatial soliton
• Solitary waves in discrete media [1]
• Topological quantum number

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. Emplit, J.P. Hamaide, F. Reynaud, C. Froehly and A. Barthelemy (1987) Picosecond steps and dark pulses through nonlinear single mode fibers. Optics. Comm. 62, 374

• P. G. Drazin and R. S. Johnson (1989). Solitons: an introduction. Cambridge University Press.

• N. Manton and P. Sutcliffe (2004). Topological solitons. Cambridge University Press.