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Solitons: Historical and Physical Introduction

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

The interest in nonlinear physics has grown significantly over the last fifty years. Although numerous nonlinear processes had been previouslyidentified the mathematic tools of nonlinear physics had not yet been developed. The available tools were linear, and nonlinearities were avoided or treatedas perturbations of linear theories. The solitary water wave,experimentally discovered in 1834 by John ScottRussell , led to numerousdiscussions. This hump-shape localized wave that propagates along one space‐direction with undeformed shape has spectacular stability properties. JohnScott Russell carried out many experiments to obtain the properties of this wave. The theories which were based on linear approaches concluded that thiskind of wave could not exist. The controversy was resolved by J. Boussinesq [5] and by LordRayleigh [64] who showed that if dissipation is neglected, the increase in local wave velocityassociated with finite amplitude is balanced by the...

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Abbreviations

Breaking waves:

As waves increase in height through the shoaling process, the crest of the wave tends tospeed up relative to the rest of the wave. Waves break when the speed of the crest exceeds the speed of the advance of the wave asa whole.

Crystal lattice:

A geometric arrangement of the points in space at which the atoms, molecules, orions of a crystal occur.

Deep water:

Water sufficiently deep that surface waves are little affected by the ocean bottom. Waterdeeper than one-half the surface wave length is considered deepwater.

Fluxon :

Quantum of magnetic flux.

Freak waves:

Single waves which result from a local focusing of wave energy. They are ofconsiderable danger to mariners because of their unexpected nature.

Geostrophic adjustment:

The process by which an unbalanced atmospheric flow field is modified togeostrophic equilibrium, generally by a mutual adjustment of the atmospheric wind and pressure fields depending on the initial horizontal scale of thedisturbance.

Geostrophic equilibrium:

A state of motion of an inviscid fluid in which the horizontal Coriolisforce exactly balances the horizontal pressure force at all points of the field.

Hydraulic jump:

A sudden turbulent rise in water level, such as often occurs at the foot ofa spillway when the velocity of rapidly flowing water is instantaneously slowed.

Katabatic wind:

Most widely used in mountain meteorology to denote a downslope flow driven bycooling at the slope surface during periods of light larger‐scale winds.

Lightning:

Lightning is a transient, high‐current electric discharge.

Plasma:

Hot, ionized gas.

Shallow water:

Water depths less than or equal to one half of the wavelength ofa wave.

Solitary wave:

Localized wave that propagates along one space direction only, with undeformedshape.

Soliton :

Large‐amplitude pulse of permanent form whose shape and speed are not altered bycollision with other solitary waves, the exact solution of a nonlinear equation.

Spillway:

A feature in a dam allowing excess water to pass without overtopping thedam.

Thermocline:

A layer in which the temperature decreases significantly (relative to the layers aboveand below) with depth.

Synoptic scale:

Used with respect to weather systems ranging in size from several hundred kilometers toseveral thousand kilometers.

Thunder:

The sound emitted by rapidly expanding gases along the channel of a lightningdischarge.

Thunderstorm:

In general, a local storm, invariably produced by a cumulonimbus cloud andalways accompanied by lightning and thunder, usually with strong gusts of wind, heavy rain, and sometimes with hail.

Tidal bore :

Tidal wave that propagates up a relatively shallow and sloping estuary or river, ina solitary wave form. The leading edge presents an abrupt rise in level, sometimes with continuous breaking and often immediately followed by severallarge undulations. The tidal bore is usually associated with high tidal range and a sharp narrowing and shoaling at the entrance. Also called pororoca(Brazilian) and mascaret (French).

Troposphere:

The portion of the atmosphere from the earth's surface to the tropopause, that is thelowest 10–20 km of the atmosphere.

Tsunami :

Long period ocean wave generated by an earthquake or a volcanicexplosion.

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Authors and Affiliations

  1. Laboratoire Ondes et Milieux Complexes, Fre CNRS 3102, Le Havre Cedex, France

    François Marin

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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Marin, F. (2009). Solitons: Historical and Physical Introduction. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_506

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