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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 delicate balance between nonlinear and dispersive effects in the medium. ("Dispersive effects" refer to a relationship 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, reproduced the phenomenon in a wave tank, and named it the "Wave of Translation". Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the branch of science concerned with the discovery and characterization of universal laws which govern matter, energy, space, and time. ...
In mathematics, nonlinear systems represent systems whose behavior is not expressible as a sum of the behaviors of its descriptors. ...
In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ...
John Scott Russell John Scott Russell (May 9, 1808, Glasgow - 8 June 1882) was a Scottish naval engineer who built The Great Eastern in collaboration with Isambard Kingdom Brunel, and made the discovery that gave birth to the modern study of solitons. ...
The Union Canal is a 50 km (31. ...
John Scott Russell John Scott Russell (May 9, 1808, Glasgow - 8 June 1882) was a Scottish naval engineer who built The Great Eastern in collaboration with Isambard Kingdom Brunel, and made the discovery that gave birth to the modern study of solitons. ...
It is not easy to define precisely what a soliton is. Drazin and Johnson (1989) describe solitons as solutions of nonlinear differential equations which
- represent waves of permanent form;
- are localised, so that they decay or approach a constant at infinity;
- 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). Waves with the same phase Waves with different phases The phase of a wave relates the position of a feature, typically a peak or a trough of the waveform, to that same feature in another part of the waveform (or, which amounts to the same, on a second waveform). ...
Nonlinear optics is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization P responds nonlinearly to the electric field E of the light. ...
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. The Kerr effect or the quadratic electro-optic effect is a change in the refractive index of a material in response to the intensity of an external electric field. ...
In optics, the term soliton is used to refer any optical field that does not change during propagation because of a delicate balance between nonlinear and linear effects in the medium. ...
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. The mathematical theory of these equations is a broad and very active field of mathematical research. In theoretical physics, an exactly solvable model or integrable model refers to a physical model, a physical theory, or set of differential equations whose exact solution may be calculated analytically in terms of elementary or special functions; the adjective integrable is therefore implies solvablility. ...
The Korteweg-de Vries equation (KdV equation for short) is the following partial differential equation for a function φ of two real variables, x and t: Its solutions clump up into solitons. ...
In theoretical physics, the nonlinear Schrödinger equation is a nonlinear version of Schrödingers equation in two dimensions. ...
The Sine-Gordon equation is a partial differential equation for a function of two real variables, x and t, given as follows: The name is a pun on the Klein-Gordon equation. ...
In mathematics, the inverse scattering transform is a procedure for integrating certain nonlinear partial differential equations (PDEs) by first converting them into a system of linear ordinary differential equations (ODEs). ...
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. The tidal bore in Upper Cook Inlet, Alaska A tidal bore (or just bore, or eagre) is a tidal phenomenon in which the leading edge of the incoming tide forms a wave (or waves) of water that travel up a river or narrow bay against the direction of the current. ...
“Severn†redirects here. ...
Internal waves are gravity waves that oscillate due to the buoyancy force. ...
For discussion of land surfaces themselves, see Terrain. ...
A pycnocline is a layer of rapid change in water density with depth. ...
The spectacular Morning Glory cloud occurs in the Australian region called the Gulf of Carpentaria and off the Mexican coast in the Sea of Cortez The springtime phenomenon is a completely natural and quite spectacular, though relatively unknown. ...
The Gulf of Carpentaria viewed from orbit. ...
Smoke rising in Lochcarron is stopped by an overlying layer of warmer air. ...
A roll cloud is a low, horizontal tube-shaped arcus cloud associated with a thunderstorm gust front (or sometimes with a cold front). ...
The Soliton model in neuroscience is a recently developed model that attempts to explain how signals are conducted within neurons. ...
Drawing of the cells in the chicken cerebellum by S. Ramón y Cajal Neuroscience is a field that is devoted to the scientific study of the nervous system. ...
Drawing by Santiago Ramón y Cajal of neurons in the pigeon cerebellum. ...
In a broader sense, solitons belong to a class of phenomomena known as "topological defects" or "topological solitons". These are a generalization of the notion of a soliton to any solution of a set of partial differential equations that is stable against decay to the "trivial solution" due to topological constraints. 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. In cosmology, a topological defect is a (often) stable configuration of matter predicted by some theories to form at phase transitions in the very early universe. ...
In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ...
In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ...
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. ...
An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ...
In materials science, a dislocation is a linear crystallographic defect, or irregularity, within a crystal structure. ...
Rose des Sables (Sand Rose), formed of gypsum crystals In mineralogy and crystallography, a crystal structure is a unique arrangement of atoms in a crystal. ...
In physics, a Dirac string is a fictitious one-dimensional curve in space, stretched from a magnetic monopole - also called the Dirac monopole - to infinity. ...
In physics, a magnetic monopole is a hypothetical particle that may be loosely described as a magnet with only one pole (see electromagnetic theory for more on magnetic poles). ...
Electromagnetism is the physics of the electromagnetic field: a field which exerts a force on particles that possess the property of electric charge, and is in turn affected by the presence and motion of those particles. ...
In theoretical physics, a skyrmion, named for Tony Skyrme, is a homotopically nontrivial classical solution of a nonlinear sigma model with a non-trivial target manifold topology i. ...
In theoretical physics and mathematics, the Wess-Zumino-Witten (WZW) model, also called the Wess-Zumino-Novikov-Witten model, is a simple model of conformal field theory whose solutions are realized by affine Kac-Moody algebras. ...
Quantum field theory (QFT) is the quantum theory of fields. ...
A cosmic string is a hypothetical 1-dimensional topological defect in the fabric of spacetime. ...
A domain wall is a theoretical 2-dimensional singularity. ...
Physical cosmology, as a branch of astrophysics, is the study of the large-scale structure of the universe and is concerned with fundamental questions about its formation and evolution. ...
History
In 1834, John Scott Russell describes his wave of translation. John Scott Russell John Scott Russell (May 9, 1808, Glasgow - 8 June 1882) was a Scottish naval engineer who built The Great Eastern in collaboration with Isambard Kingdom Brunel, and made the discovery that gave birth to the modern study of solitons. ...
John Scott Russell John Scott Russell (May 9, 1808, Glasgow - 8 June 1882) was a Scottish naval engineer who built The Great Eastern in collaboration with Isambard Kingdom Brunel, and made the discovery that gave birth to the modern study of solitons. ...
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. Bell Laboratories (also known as Bell Labs and formerly known as AT&T Bell Laboratories and Bell Telephone Laboratories) was the main research and development arm of the United States Bell System. ...
Martin David Kruskal (b. ...
Princeton University is a private coeducational research university located in Princeton, New Jersey, in the United States of America. ...
The Korteweg-de Vries equation (KdV equation for short) is the following partial differential equation for a function φ of two real variables, x and t: Its solutions clump up into solitons. ...
porc A finite difference is a mathematical expression of the form f(x + b) − f(x +a). ...
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. In mathematics, the inverse scattering transform is a procedure for integrating certain nonlinear partial differential equations (PDEs) by first converting them into a system of linear ordinary differential equations (ODEs). ...
In mathematics, an analytic function is a function that is locally given by a convergent power series. ...
Peter David Lax (born May 1,1926) is a highly-respected mathematician working in the areas of mathematics. ...
In mathematics, in the theory of differential equations, a Lax pair is a pair of time-dependent matrices that describe certain solutions of differential equations. ...
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. AT&T Inc. ...
Bell Laboratories (also known as Bell Labs and formerly known as AT&T Bell Laboratories and Bell Telephone Laboratories) was the main research and development arm of the United States Bell System. ...
Optical fibers An optical fiber (or fibre) is a glass or plastic fiber designed to guide light along its length by confining as much light as possible in a propagating form. ...
Self-phase modulation (SPM) is a nonlinear optical effect of light-matter interaction. ...
Dispersion of a light beam in a prism. ...
Telecommunication involves the transmission of signals over a distance for the purpose of communication. ...
Solitons in a fiber optic system are described by the Manakov equations. Maxwells Equations, when converted to cylindrical coordinates, and with the boundary conditions for a fiber optic cable while including birefringence as an effect taken into account, will yield the coupled nonlinear Schrödinger 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. When light is scattered from a molecule most photons are elastically scattered. ...
Sir Chandrasekhara Venkata Raman, CBE (Tamil: சநà¯à®¤à®¿à®°à®šà¯‡à®•à®° வெஙà¯à®•à®Ÿà®°à®¾à®®à®©à¯) (November 7, 1888 – November 21, 1970) was an Indian physicist, who was awarded the 1930 Nobel Prize in Physics for his work on the scattering of light and for the discovery of the Raman effect, which is named after him. ...
The 1920s is a decade that is sometimes referred to as the Jazz Age or the Roaring Twenties, usually applied to America. ...
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. General Name, Symbol, Number erbium, Er, 68 Chemical series lanthanides Group, Period, Block n/a, 6, f Appearance silvery white Standard atomic weight 167. ...
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). France Télécom (Euronext: FTE, NYSE: FTE) (often spelled France Telecom, without the accents, in non-French text) is the main telecommunication company in France. ...
In telecommunications wavelength division multiplexing (WDM) is a technology which multiplexes several optical carrier signals on a single optical fibre by using different wavelengths (colours) of laser light to carry different signals. ...
In computing, binary prefixes can be used to quantify large numbers where powers of two are more useful than powers of ten. ...
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. Telecommunication involves the transmission of signals over a distance for the purpose of communication. ...
John Scott Russell John Scott Russell (May 9, 1808, Glasgow - 8 June 1882) was a Scottish naval engineer who built The Great Eastern in collaboration with Isambard Kingdom Brunel, and made the discovery that gave birth to the modern study of solitons. ...
Bions The bound state of two solitons is know as a bion. Image File history File links This is a lossless scalable vector image. ...
See also In optics, the term soliton is used to refer any optical field that does not change during propagation because of a delicate balance between nonlinear and linear effects in the medium. ...
A topological soliton is a solution of a system of partial differential equations (or alternatively, a quantum field theory), not so much because of the nature of the PDEs themselves, but because of the boundary conditions entailing the existence of homotopically distinct solutions. ...
A solitary wave is a special sort of solution of a non-linear partial differential equation. ...
The Draupner wave, a single giant wave measured on New Years Day 1995, finally confirmed the existence of freak waves, which had previously been considered near-mythical Freak waves, also known as rogue waves or monster waves, are relatively large and spontaneous ocean surface waves which can sink even...
It has been suggested that this article or section be merged with Soliton (topological). ...
It has been suggested that this article or section be merged with breather. ...
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.
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