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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. This article describes the use of the term nonlinearity in mathematics "Dispersive effects" refer to dispersion relations between the frequency and the speed of the waves. Dispersion relations describe the ways that wave propagation varies with the Wavelength or Frequency of a wave. Solitons arise as the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i 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). John Scott Russell ( 9 May 1808, Parkhead, Glasgow – 8 June 1882, Ventor, Isle of Wight) was a The Union Canal is a 315 mile (507 km Contour canal in Scotland, from Lochrin Basin Fountainbridge, Edinburgh to Falkirk He reproduced the phenomenon in a wave tank and named it the "Wave of Translation". John Scott Russell ( 9 May 1808, Parkhead, Glasgow – 8 June 1882, Ventor, Isle of Wight) was a

Contents 1 Definition 2 Explanation 3 History 4 Solitons in fiber optics 5 Solitons in magnets 6 Bions 7 See also 8 References 9 External links

Definition

A single, consensus definition of a soliton is difficult to find. 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. The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0

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). Nonlinear optics (NLO is the branch of Optics that describes the behaviour of Light in nonlinear media, that is media in which the dielectric polarization

Explanation

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. The Kerr effect or the quadratic electro-optic effect ( QEO effect) is a change in the Refractive index of a material in response to an Electric field If the pulse has just the right shape, the Kerr effect will exactly cancel the dispersion effect, and the pulse's shape won't change over time: a soliton. See soliton (optics) for a much more detailed description. In Optics, the term soliton is used to refer to any Optical field that does not change during propagation because of a delicate balance between nonlinear

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. In Mathematics and Physics, there are various distinct notions that are referred to under the name of integrable systems. In Mathematics, the Korteweg–de Vries equation ( KdV equation for short is a Mathematical model of waves on shallow water surfaces In Theoretical physics, the nonlinear Schrödinger equation (NLS is a nonlinear version of Schrödinger's equation. The sine-Gordon equation is a nonlinear hyperbolic Partial differential equation in 1+1 dimensions involving the D'Alembert operator and the sine of the The soliton solutions are typically obtained by means of the inverse scattering transform and owe their stability to the integrability of the field equations. In Mathematics, the inverse scattering transform is a method for solving some non-linear Partial differential 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. 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 For other rivers named "Severn" see Severn River. The River Severn ( Welsh: Afon Hafren, Latin Other solitons occur as the undersea internal waves, initiated by seabed topography, that propagate on the oceanic pycnocline. Internal waves are Gravity waves that oscillate within rather than on the surface of a fluid medium Topography ( topo-, "place" and graphia, "writing" is the study of Earth 's Surface features or those of Planets A pycnocline is a rapid change in water Density with depth. In Freshwater environments such as Lakes this density change is primarily caused 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 Morning Glory cloud is a rare meteorological phenomenon observed in Northern Australia 's Gulf of Carpentaria. The Gulf of Carpentaria is a large shallow sea enclosed on three sides by northern Australia and bounded on the north by the Arafura Sea (the body of water that lies In meteorology an inversion is a deviation from the normal change of an atmospheric property with altitude The recent and not widely accepted soliton model in neuroscience proposes to explain the signal conduction within neurons as pressure solitons. The Soliton model in Neuroscience is a recently developed model that attempts to explain how signals are conducted within Neurons It proposes that the signals Neuroscience is a field devoted to the scientific study of the nervous system Neurons (ˈnjuːɹɒn also known as neurones and nerve cells) are responsive cells in the Nervous system that process and transmit information

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. Also see base concepts Topology, Differential equations Quantum theory & Condensed matter physics. Also see base concepts Topology, Differential equations Quantum theory & Condensed matter physics. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i " Soliton stability is due to topological constraints, rather than integrability of the field equations. The constraints arise 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. In Mathematics, in the field of Differential equations a boundary value problem is a Differential equation together with a set of additional restraints In Mathematics, homotopy groups are used in Algebraic topology to classify Topological spaces The base point preserving maps from an n -dimensional Thus, the differential equation solutions can be classified into homotopy classes. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical There is no continuous transformation that will map a solution in one homotopy class to another. 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 Materials science, a dislocation is a Crystallographic defect, or irregularity within a Crystal structure. 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 conceived of by the physicist Dirac, stretched from a Magnetic monopole - also In Physics, a magnetic monopole is a hypothetical particle that is a Magnet with only one pole (see Maxwell's equations for more on magnetic Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of In Theoretical physics, a skyrmion, conceived by Tony Skyrme, is a mathematical model used to model Baryons (a Subatomic particle) In Theoretical physics and Mathematics, the Wess-Zumino-Witten (WZW model, also called the Wess-Zumino-Novikov-Witten model, is a simple model of In quantum field theory (QFT the forces between particles are mediated by other particles A cosmic string is a hypothetical 1-dimensional (spatially Topological defect in various fields A domain wall is a term used in Physics which can have one of two distinct but similar meanings in either Magnetism or String theory. Physical cosmology, as a branch of Astronomy, is the study of the large-scale structure of the Universe and is concerned with fundamental questions about its

History

In 1834, John Scott Russell describes his wave of translation. John Scott Russell ( 9 May 1808, Parkhead, Glasgow – 8 June 1882, Ventor, Isle of Wight) was a John Scott Russell ( 9 May 1808, Parkhead, Glasgow – 8 June 1882, Ventor, Isle of Wight) was a The discovery is described here in Russell's own words:

"I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped - not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation". ^[1]

(Note: This passage has been repeated in many papers and books on soliton theory. )

(Note: "Translation" here means that there is real mass transport such that water can be transported from one end of the canal to the other end by this "Wave of Translation". Usually there is no real mass transport from one side to another side for ordinary waves. )

Russell spent some time making practical and theoretical investigations of these waves, he built wave tanks at his home and noticed some key properties:

• The waves are stable, and can travel over very large distances (normal waves would tend to either flatten out, or steepen and topple over)
• The speed depends on the size of the wave, and its width on the depth of water.
• Unlike normal waves they will never merge — so a small wave is overtaken by a large one, rather than the two combining.
• If a wave is too big for the depth of water, it splits into two, one big and one small.

Russell's experimental work seemed at odds with the Isaac Newton and Daniel Bernoulli's theories of hydrodynamics. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements Daniel Bernoulli ( Groningen, 29 January 1700 &ndash 27 July 1782 was a Dutch - Swiss Mathematician, who is particularly remembered for his applications Fluid dynamics is the sub-discipline of Fluid mechanics dealing with fluid flow: Fluids ( Liquids and Gases in motion George Biddell Airy and George Gabriel Stokes had difficulty accepting Russell's experimental observations because they could not be explained by linear water wave theory. Sir George Biddell Airy FRS (27 July 1801&ndash2 January 1892 was an English Mathematician and Astronomer, Astronomer Royal Sir George Gabriel Stokes 1st Baronet FRS ( 13 August 1819 &ndash 1 February 1903) was a mathematician and physicist His contemporaries spent some time attempting to extend the theory but it would take until 1895 before Diederik Korteweg and Gustav de Vries provided the theoretical explanation. Year 1895 ( MDCCCXCV) was a Common year starting on Tuesday (link will display full calendar of the Gregorian calendar (or a Common year Diederik Johannes Korteweg ( 31 March 1848, Den Bosch – 10 May 1941, Amsterdam) was a Dutch Mathematician Gustav de Vries ( January 22, 1866 &ndash December 16, 1934) was a Dutch mathematician who is best remembered for his work on the Korteweg–de ^[2]

(Note: Lord Rayleigh published a paper in Philosophical Magazine in 1876 to support John Scott Russell's experimental observation with his mathematical theory. John William Strutt 3rd Baron Rayleigh OM (12 November 1842 &ndash 30 June 1919 was an English Physicist who with William Ramsay, discovered In his 1876 paper, Lord Rayleigh mentioned Russell's name and also admitted that the first theoretical treatment was by Joseph Valentin Boussinesq in 1871. Joseph Boussinesq mentioned Russell's name in his 1871 paper. Joseph Valentin Boussinesq (born March 13 1842 in Saint-André-de-Sangonis ( Hérault département) died February Thus Russell's observations on solitons were accepted as true by some prominent scientists within his own life time of 1808-1882. Korteweg and de Vries did not mention John Scott Russell's name at all in their 1895 paper but they did quote Boussinesq's paper in 1871 and Lord Rayleigh's paper in 1876. The paper by Korteweg and de Vries in 1895 was not the first theoretical treatment of this subject but it was a very important milestone in the history of the development of soliton theory. )

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) is the Research organization Martin David Kruskal ( September 28 1925 &ndash December 26 2006) was an American Mathematician and Physicist. Princeton University is a private Coeducational research university located in Princeton, New Jersey. In Mathematics, the Korteweg–de Vries equation ( KdV equation for short is a Mathematical model of waves on shallow water surfaces A finite difference is a mathematical expression of the form f ( x + b) &minus f ( x + a)

In 1967, Gardner, Greene, Kruskal and Miura discovered an inverse scattering transform enabling analytical solution of the KdV equation. In Mathematics, the inverse scattering transform is a method for solving some non-linear Partial differential equations. This article is about both real and complex analytic functions The work of Peter Lax on Lax pairs and the Lax equation has since extended this to solution of many related soliton-generating systems. Peter David Lax (born May 1, 1926 in Budapest, Hungary) is a Mathematician working in the areas of pure and applied 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

Much experimentation has been done using solitons in fiber optics applications. Solitons' inherent stability make long-distance transmission possible without the use of repeaters, and could potentially double transmission capacity as well. A repeater is an electronic device that receives a signal and Retransmits it at a higher level and/or higher power or onto the other side of an obstruction ^[3]

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. Before proposing a merge request please see Talk and see if the merger you propose has recently been made and Bell Laboratories (also known as Bell Labs and formerly known as AT&T Bell Laboratories and Bell Telephone Laboratories) is the Research organization An optical fiber (or fibre) is a Glass or Plastic fiber that carries Light along its length Self-phase modulation (SPM is a nonlinear optical effect of Light - Matter interaction In Optics, dispersion is the phenomenon in which the Phase velocity of a wave depends on its frequency 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. Maxwell's Equations, when converted to Cylindrical coordinates, and with the boundary conditions for an Optical fiber while including Birefringence as an

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. Raman scattering or the Raman effect (pronounced — is the inelastic scattering of a Photon. Sir Chandrasekhara Venkata Raman, FRS (சந்திரசேகர வெங்கடராமன ( 7 November 1888 &ndash 21 November The 1920s is sometimes referred to as the " Jazz Age " or the " Roaring Twenties " when speaking about the United States and Canada

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). Erbium (ˈɝbiəm is a Chemical element with the symbol Er and Atomic number 68 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). France Télécom () is the main Telecommunication company in France and one of the largest in the world In Fiber-optic communications wavelength-division multiplexing ( WDM) is a technology which multiplexes multiple optical carrier signals on a In computing binary prefixes are names or associated symbols that can precede a unit of measure (such as a Byte) to indicate multiplication by a power of two

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 waves.

Solitons in magnets

In magnets, there also exist different types solitons and other nonlinear waves. These magnetic solitons are an exact solutions of classical nonlinear differential equations - magnetic equations, e. g. the Landau-Lifshitz equation, continuum Heisenberg model, Ishimori equation, Mikhailov-Yaremchuk equation, nonlinear Schrodinger equation and so on. In physics the Landau-Lifshitz equation ( LLE) named for Lev Landau and Evgeny Lifshitz, is a name used for several different differential equations The Ishimori equation (IE is a Partial differential equation proposed by the Japanese Mathematician. In Theoretical physics, the nonlinear Schrödinger equation (NLS is a nonlinear version of Schrödinger's equation.

Bions

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

In field theory Bion usually refers to the solution of the Born-Infeld model. In physics it is a particular example of what is usually known as a Nonlinear electrodynamics. The name appears to have been coined by G. W. Gibbons in order to distinguish this solution from the conventional soliton, understood as a regular, finite-energy (and usually stable) solution of a differential equation describing some physical system. The word regular means a smooth solution carrying no sources at all. However, the solution of the Born-Infeld model still carries a source in the form of a Dirac-delta function at the origin. As a consequence it displays a singularity in this point (although the electric field is everywhere regular). In some physical contexts (for instance string theory) this feature can be important, which motivated the introduction of a special name for this class of solitons.

On the other hand, when gravity is added (i. e. when considering the coupling of the Born-Infeld model to General Relativity) the corresponding solution is called EBIon, where "E" stands for "Einstein".

See also

• Clapotis
• Freak waves may be a related phenomenon. In Hydrodynamics, the clapotis (from "lapping of water" is a non-breaking Standing wave pattern caused for example by the reflection of a traveling surface Rogue waves, also known as freak waves, monster waves or extreme waves, are relatively large and spontaneous Ocean surface waves that are a threat
• Oscillons
• Q-Ball a non-topological soliton
• Soliton (topological). In Physics, an oscillon is a Soliton -like phenomenon that results from vibrating a plate with a large number of small uniform particles placed freely on top For the ball used in Billiards, see Cue ball. "Q ball" is also the name of the Q sensor module in the Apollo spacecraft Also see base concepts Topology, Differential equations Quantum theory & Condensed matter physics.
• 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.

• Linn F. Mollenauer and James P. Gordon (2006). Solitons in optical fibers. Elsevier Academic Press.

• R. Rajaraman (1982). Solitons and instantons. North-Holland.

External links

• Solitons, solitary waves and secondary or baby solitary waves in discrete media
• Heriot-Watt University soliton page
• The many faces of solitons
• Klaus Brauer's soliton page
• Solitons and Soliton Collisions
• John Scott Russell and the solitary wave
• Severn Bore web site
• John Scott Russell biography
• Soliton in Electrical Engineering
• Miura's home page
• Photograph of Soliton on the Scott Russell Aqueduct
• Solitons possible agent of nerve transmission (PDF)(pnas.org)
• Three Solitons Solution of KdV Equation
• Three Solitons (unstable) Solution of KdV Equation
• Solitons & Nonlinear Wave Equations

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