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Solitons: Historical and Physical Introduction

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Mathematics of Complexity and Dynamical Systems

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Article Outline

Glossary

Definition of the Subject

Introduction

Historical Discovery of Solitons

Physical Properties of Solitons and Associated Applications

Mathematical Methods Suitable for the Study of Solitons

Future Directions

Bibliography

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Abbreviations

Breaking waves:

As waves increase in height through the shoaling process, the crest of the wave tends to speed 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 as a whole.

Crystal lattice:

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

Deep water:

Water sufficiently deep that surface waves are little affected by the ocean bottom. Water deeper 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 of considerable danger to mariners because of their unexpected nature.

Geostrophic adjustment:

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

Geostrophic equilibrium:

A state of motion of an inviscid fluid in which the horizontal Coriolis force 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 of a 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 by cooling 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 of a wave.

Solitary wave:

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

Soliton :

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

Spillway:

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

Thermocline:

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

Synoptic scale:

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

Thunder:

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

Thunderstorm:

In general, a local storm, invariably produced by a cumulonimbus cloud and always 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, in a solitary wave form. The leading edge presents an abrupt rise in level, sometimes with continuous breaking and often immediately followed by several large 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 the lowest 10–20 km of the atmosphere.

Tsunami :

Long period ocean wave generated by an earthquake or a volcanic explosion.

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  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. (2012). Solitons: Historical and Physical Introduction. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_100

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