In

mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

and

physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, a

**soliton** is a self-reinforcing solitary

waveIn physics, a wave is a disturbance that travels through space and time, accompanied by the transfer of energy.Waves travel and the wave motion transfers energy from one point to another, often with no permanent displacement of the particles of the medium—that is, with little or no associated mass...

(a

wave packetIn physics, a wave packet is a short "burst" or "envelope" of wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of different wavenumbers, with phases and amplitudes such that they interfere...

or pulse) that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of

nonlinearIn mathematics, a nonlinear system is one that does not satisfy the superposition principle, or one whose output is not directly proportional to its input; a linear system fulfills these conditions. In other words, a nonlinear system is any problem where the variable to be solved for cannot be...

and

dispersive effectsIn physics and electrical engineering, dispersion most often refers to frequency-dependent effects in wave propagation. Note, however, that there are several other uses of the word "dispersion" in the physical sciences....

in the medium. (The term "dispersive effects" refers to a property of certain systems where the speed of the waves varies according to frequency.) Solitons arise as the solutions of a widespread class of weakly nonlinear dispersive

partial differential equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s describing physical systems. The soliton phenomenon was first described by

John Scott RussellJohn Scott Russell 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.-Personal life:John Scott Russell was born John Russell on 9 May 1808 in Parkhead, Glasgow, the son of...

(1808–1882) who observed a solitary wave in the

Union CanalThe Union Canal is a 31.5-mile canal in Scotland, from Lochrin Basin, Fountainbridge, Edinburgh to Falkirk, where it meets the Forth and Clyde Canal.-Location and features:...

in Scotland. He reproduced the phenomenon in a

wave tankA wave tank is a laboratory setup for observing the behavior of surface waves. The typical wave tank is a deep and narrow, transparent-sided box filled with liquid, usually water, leaving open or air-filled space on top. At one end of the tank an actuator generates waves; the other end usually...

and named it the "Wave of Translation".

## Definition

A single, consensus definition of a soliton is difficult to find. ascribe three properties to solitons:

- They are of permanent form;
- They are localised within a region;
- They can interact with other solitons, and emerge from the collision unchanged, except for a phase shift
Phase in waves is the fraction of a wave cycle which has elapsed relative to an arbitrary point.-Formula:The phase of an oscillation or wave refers to a sinusoidal function such as the following:...

.

More formal definitions exist, but they require substantial mathematics. Moreover, some scientists use the term

*soliton* for phenomena that do not quite have these three properties (for instance, the 'light bullets' of

nonlinear opticsNonlinear optics is the branch of optics that describes the behavior of light in nonlinear media, that is, media in which the dielectric polarization P responds nonlinearly to the electric field E of the light...

are often called solitons despite losing energy during interaction).

## 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 effectThe Kerr effect, also called the quadratic electro-optic effect , is a change in the refractive index of a material in response to an applied electric field. The Kerr effect is distinct from the Pockels effect in that the induced index change is directly proportional to the square of the electric...

: the

refractive indexIn optics the refractive index or index of refraction of a substance or medium is a measure of the speed of light in that medium. It is expressed as a ratio of the speed of light in vacuum relative to that in the considered medium....

of a material at 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 dispersion effect, and the pulse's shape won't change over time: a soliton. See

soliton (optics)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 and linear effects in the medium. There are two main kinds of solitons:...

for a more detailed description.

Many exactly solvable models have soliton solutions, including the

Korteweg–de Vries equationIn mathematics, the Korteweg–de Vries equation is a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an exactly solvable model, that is, a non-linear partial differential equation whose solutions can be exactly and precisely specified...

, the

nonlinear Schrödinger equationIn theoretical physics, the nonlinear Schrödinger equation is a nonlinear version of Schrödinger's equation. It is a classical field equation with applications to optics and water waves. Unlike the Schrödinger equation, it never describes the time evolution of a quantum state...

, the coupled nonlinear Schrödinger equation, and the

sine-Gordon equationThe sine–Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally considered in the nineteenth century in the course of study of surfaces of constant negative...

. The soliton solutions are typically obtained by means of the

inverse scattering transformIn mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. It is one of the most important developments in mathematical physics in the past 40 years...

and owe their stability to the

integrabilityIntegrability may refer to:* Riemann integrability; see Riemann integral* Lebesgue integrability; see Lebesgue integral* Darboux integrability; see Darboux integral* System integration * Interoperability...

of the field equations. The mathematical theory of these equations is a broad and very active field of mathematical research.

Some types of

tidal boreA tidal bore is a tidal phenomenon in which the leading edge of the incoming tide forms a wave of water that travel up a river or narrow bay against the direction of the river or bay's current...

, a wave phenomenon of a few rivers including the

River SevernThe River Severn is the longest river in Great Britain, at about , but the second longest on the British Isles, behind the River Shannon. It rises at an altitude of on Plynlimon, Ceredigion near Llanidloes, Powys, in the Cambrian Mountains of mid Wales...

, are 'undular': a wavefront followed by a train of solitons. Other solitons occur as the undersea

internal waveInternal waves are gravity waves that oscillate within, rather than on the surface of, a fluid medium. They are one of many types of wave motion in stratified fluids . A simple example is a wave propagating on the interface between two fluids of different densities, such as oil and water...

s, initiated by seabed

topographyTopography is the study of Earth's surface shape and features or those ofplanets, moons, and asteroids...

, that propagate on the oceanic

pycnoclineA pycnocline is the cline or layer where the density gradient is greatest within a body of water. An ocean current is generated by the forces such as breaking waves, terms of temperature and salinity differences, wind, Coriolis effect, and tides caused by the gravitational pull of the Moon and the...

. Atmospheric solitons also exist, such as the

Morning Glory CloudThe Morning Glory cloud is a rare meteorological phenomenon occasionally observed in different locations around the world. The southern part of Northern Australia's Gulf of Carpentaria is the only known location where it can be predicted and observed on a more or less regular basis. The settlement...

of the

Gulf of CarpentariaThe Gulf of Carpentaria is a large, shallow sea enclosed on three sides by northern Australia and bounded on the north by the Arafura Sea...

, where pressure solitons travelling in a temperature inversion layer produce vast linear roll clouds. The recent and not widely accepted

soliton modelThe Soliton model in neuroscience is a recently developed model that attempts to explain how signals are conducted within neurons. It proposes that the signals travel along the cell's membrane in the form of certain kinds of sound pulses known as solitons...

in

neuroscienceNeuroscience is the scientific study of the nervous system. Traditionally, neuroscience has been seen as a branch of biology. However, it is currently an interdisciplinary science that collaborates with other fields such as chemistry, computer science, engineering, linguistics, mathematics,...

proposes to explain the signal conduction within

neuronA neuron is an electrically excitable cell that processes and transmits information by electrical and chemical signaling. Chemical signaling occurs via synapses, specialized connections with other cells. Neurons connect to each other to form networks. Neurons are the core components of the nervous...

s as pressure solitons.

A topological soliton, also called a topological defect, is any solution of a set of

partial differential equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

s that is stable against decay to the "trivial solution." Soliton stability is due to topological constraints, rather than

integrabilityIntegrability may refer to:* Riemann integrability; see Riemann integral* Lebesgue integrability; see Lebesgue integral* Darboux integrability; see Darboux integral* System integration * Interoperability...

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 groupIn mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...

, preserved by the differential equations. Thus, the differential equation solutions can be classified into homotopy classes. 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 stringIn physics, a Dirac string is a fictitious one-dimensional curve in space, conceived of by the physicist Paul Dirac, stretching between two Dirac magnetic monopoles with opposite magnetic charges, or from one magnetic monopole out to infinity. The gauge potential cannot be defined on the Dirac...

and the

magnetic monopoleA magnetic monopole is a hypothetical particle in particle physics that is a magnet with only one magnetic pole . In more technical terms, a magnetic monopole would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring...

in

electromagnetismElectromagnetism is one of the four fundamental interactions in nature. The other three are the strong interaction, the weak interaction and gravitation...

, the

SkyrmionIn theoretical physics, a skyrmion is a mathematical model used to model baryons . It was conceived by Tony Skyrme.-Overview:...

and the Wess–Zumino–Witten model in

quantum field theoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

, and

cosmic stringCosmic strings are hypothetical 1-dimensional topological defects which may have formed during a symmetry breaking phase transition in the early universe when the topology of the vacuum manifold associated to this symmetry breaking is not simply connected. It is expected that at least one string...

s and

domain wallA domain wall is a term used in physics which can have one of two distinct but similar meanings in magnetism, optics, or string theory. These phenomena can all be generically described as topological solitons which occur whenever a discrete symmetry is spontaneously broken.-Magnetism:In magnetism,...

s in

cosmologyPhysical cosmology, as a branch of astronomy, is the study of the largest-scale structures and dynamics of the universe and is concerned with fundamental questions about its formation and evolution. For most of human history, it was a branch of metaphysics and religion...

.

## History

In 1834,

John Scott RussellJohn Scott Russell 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.-Personal life:John Scott Russell was born John Russell on 9 May 1808 in Parkhead, Glasgow, the son of...

describes his

*wave of translation*.

["Translation" here means that there is real mass transport, although it is not the same water which is transported from one end of the canal to the other end by this "Wave of Translation". Rather, a fluid parcel acquires momentum]In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

during the passage of the solitary wave, and comes to rest again after the passage of the wave. But the fluid parcel has been displaced substantially forward during the process – by Stokes driftFor a pure wave motion in fluid dynamics, the Stokes drift velocity is the average velocity when following a specific fluid parcel as it travels with the fluid flow...

in the wave propagation direction. And a net mass transport is the result. Usually there is little mass transport from one side to another side for ordinary waves. The discovery is described here in Scott Russell's own words:

[This passage has been repeated in many papers and books on soliton theory.]
"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".

Scott 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.

Scott Russell's experimental work seemed at odds with

Isaac NewtonSir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

's and

Daniel BernoulliDaniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...

's theories of hydrodynamics.

George Biddell AirySir George Biddell Airy PRS KCB was an English mathematician and astronomer, Astronomer Royal from 1835 to 1881...

and

George Gabriel StokesSir George Gabriel Stokes, 1st Baronet FRS , was an Irish mathematician and physicist, who at Cambridge made important contributions to fluid dynamics , optics, and mathematical physics...

had difficulty accepting Scott Russell's experimental observations because they could not be explained by the existing water wave theories. Their contemporaries spent some time attempting to extend the theory but it would take until the 1870s before Joseph Boussinesq and Lord Rayleigh published a theoretical treatment and solutions.

[Lord Rayleigh published a paper in Philosophical Magazine in 1876 to support John Scott Russell's experimental observation with his mathematical theory. In his 1876 paper, Lord Rayleigh mentioned Scott 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. Thus Scott Russell's observations on solitons were accepted as true by some prominent scientists within his own lifetime of 1808–1882.] In 1895

Diederik KortewegDiederik Johannes Korteweg was a Dutch mathematician. He is now remembered as the joint discoverer of the Korteweg–de Vries equation.-Early life and education:...

and

Gustav de VriesGustav de Vries was a Dutch mathematician, who is best remembered for his work on the Korteweg–de Vries equation with Diederik Korteweg. He was born on 22 January 1866 in Amsterdam, and studied at the University of Amsterdam with the distinguished physical chemist Johannes van der Waals and with...

provided what is now known as the

Korteweg–de Vries equationIn mathematics, the Korteweg–de Vries equation is a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an exactly solvable model, that is, a non-linear partial differential equation whose solutions can be exactly and precisely specified...

, including solitary wave and periodic

cnoidal waveIn fluid dynamics, a cnoidal wave is a nonlinear and exact periodic wave solution of the Korteweg–de Vries equation. These solutions are in terms of the Jacobi elliptic function cn, which is why they are coined cnoidal waves...

solutions.

[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 of 1871 and Lord Rayleigh's paper of 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 ZabuskyNorman J. Zabusky is an American physicist, who is noted for the discovery of the soliton in the Korteweg–de Vries equation, in work completed with Martin Kruskal. This result early in his career was followed by an extensive body of work in computational fluid dynamics, which led him more...

of

Bell LabsBell Laboratories is the research and development subsidiary of the French-owned Alcatel-Lucent and previously of the American Telephone & Telegraph Company , half-owned through its Western Electric manufacturing subsidiary.Bell Laboratories operates its...

and

Martin KruskalMartin David Kruskal was an American mathematician and physicist. He made fundamental contributions in many areas of mathematics and science, ranging from plasma physics to general relativity and from nonlinear analysis to asymptotic analysis...

of

Princeton UniversityPrinceton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....

first demonstrated soliton behaviour in media subject to the

Korteweg–de Vries equationIn mathematics, the Korteweg–de Vries equation is a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an exactly solvable model, that is, a non-linear partial differential equation whose solutions can be exactly and precisely specified...

(KdV equation) in a computational investigation using a

finite differenceA finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient...

approach. They also showed how this behavior explained the puzzling earlier work of Fermi, Pasta and Ulam.

In 1967, Gardner, Greene, Kruskal and Miura discovered an

inverse scattering transformIn mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. It is one of the most important developments in mathematical physics in the past 40 years...

enabling

analyticalIn mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...

solution of the KdV equation. The work of

Peter LaxPeter David Lax is a mathematician working in the areas of pure and applied mathematics. He has made important contributions to integrable systems, fluid dynamics and shock waves, solitonic physics, hyperbolic conservation laws, and mathematical and scientific computing, among other fields...

on

Lax pairIn mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that describe the corresponding differential equations. They were introduced by Peter Lax to discuss solitons in continuous media...

s and the Lax equation has since extended this to solution of many related soliton-generating systems.

## Solitons in fiber optics

Much experimentation has been done using solitons in fiber optics applications. Solitons in a fiber optic system are described by the Manakov equations.

Solitons' inherent stability make long-distance transmission possible without the use of

repeaterA repeater is an electronic device that receives asignal and retransmits it at a higher level and/or higher power, or onto the other side of an obstruction, so that the signal can cover longer distances.-Description:...

s, and could potentially double transmission capacity as well.

Year | Discovery |
---|

1973 |
Akira Hasegawa of AT&TAT&T Inc. is an American multinational telecommunications corporation headquartered in Whitacre Tower, Dallas, Texas, United States. It is the largest provider of mobile telephony and fixed telephony in the United States, and is also a provider of broadband and subscription television services... Bell LabsBell Laboratories is the research and development subsidiary of the French-owned Alcatel-Lucent and previously of the American Telephone & Telegraph Company , half-owned through its Western Electric manufacturing subsidiary.Bell Laboratories operates its... was the first to suggest that solitons could exist in optical fiberAn optical fiber is a flexible, transparent fiber made of a pure glass not much wider than a human hair. It functions as a waveguide, or "light pipe", to transmit light between the two ends of the fiber. The field of applied science and engineering concerned with the design and application of... s, due to a balance between self-phase modulationSelf-phase modulation is a nonlinear optical effect of light-matter interaction.An ultrashort pulse of light, when travelling in a medium, will induce a varying refractive index of the medium due to the optical Kerr effect... and anomalous dispersionIn optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency, or alternatively when the group velocity depends on the frequency.Media having such a property are termed dispersive media... . Also in 1973 Robin BulloughRobin K. Bullough was a British Mathematical Physicist famous for his contributions to the theory of solitons, in particular for his role in the development of the theory of the optical soliton, now commonly used, for example, in the theory of trans-oceanic optical fibre communication theory, but... made the first mathematical report of the existence of optical solitons. He also proposed the idea of a soliton-based transmission system to increase performance of optical telecommunicationTelecommunication is the transmission of information over significant distances to communicate. In earlier times, telecommunications involved the use of visual signals, such as beacons, smoke signals, semaphore telegraphs, signal flags, and optical heliographs, or audio messages via coded... s. |

1987 |
– from the Universities of Brussels and Limoges – made the first experimental observation of the propagation of a dark soliton, in an optical fiber. |

1988 |
Linn Mollenauer and his team transmitted soliton pulses over 4,000 kilometers using a phenomenon called the Raman effect, named after Sir C. V. RamanSir Chandrasekhara Venkata Raman, FRS was an Indian physicist whose work was influential in the growth of science in the world. He was the recipient of the Nobel Prize for Physics in 1930 for the discovery that when light traverses a transparent material, some of the light that is deflected... who first described it in the 1920s, to provide optical gain in the fiber. |

1991 |
A Bell Labs research team transmitted solitons error-free at 2.5 gigabits per second over more than 14,000 kilometers, using erbium Erbium is a chemical element in the lanthanide series, with the symbol Er and atomic number 68. A silvery-white solid metal when artificially isolated, natural erbium is always found in chemical combination with other elements on Earth... 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. |

1998 |
Thierry Georges and his team at France TelecomFrance Telecom S.A. is the main telecommunications company in France, the third-largest in Europe and one of the largest in the world. It currently employs about 180,000 people and has 192.7 million customers worldwide . In 2010 the group had revenue of €45.5 billion... R&D Center, combining optical solitons of different wavelengthIn physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a... s (wavelength-division multiplexingIn fiber-optic communications, wavelength-division multiplexing is a technology which multiplexes a number of optical carrier signals onto a single optical fiber by using different wavelengths of laser light... ), demonstrated a *composite* data transmission of 1 terabitIn computing, a binary prefix is a specifier or mnemonic that is prepended to the units of digital information, the bit and the byte, to indicate multiplication by a power of 2... per second (1,000,000,000,000 units of information per second), not to be confused with Terabit-Ethernet.
The above impressive experiments have not translated to actual commercial soliton system deployments however, in either terrestrial or submarine systems, chiefly due to the Gordon–Haus (GH) jitter. The GH jitter requires sophisticated, expensive compensatory solutions that ultimately makes DWDM soliton transmission in the field unattractive, compared to the conventional non-return-to-zero/return-to-zero paradigm. Further, the likely future adoption of the more spectrally efficient phase-shift-keyed/QAM formats makes soliton transmission even less viable, due to the Gordon-Mollenauer effect. Consequently, the long-haul fiberoptic transmission soliton has remained a laboratory curiosity. |

2000 |
Cundiff predicted the existence of a vector soliton In physical optics or wave optics, a vector soliton is a solitary wave with multiple components coupled together that maintains its shape during propagation. Ordinary solitons maintain their shape but have effectively only one polarization component, while vector solitons have two distinct... in a birefringence fiber cavity passively mode locking through SESAMSesam was a Scandinavian internet search engine developed by the media corporation Schibsted. It was available both in a Norwegian and Swedish version and was launched on 1 November 2005. By 2007 Sesam.no had 480,000 unique users and was among the 12 largest web sites in Norway... . The polarization state of such a vector soliton could either be rotating or locked depending on the cavity parameters. |

2008 |
D. Y. Tang *et al.* observed a novel form of higher-order vector solitonIn physical optics or wave optics, a vector soliton is a solitary wave with multiple components coupled together that maintains its shape during propagation. Ordinary solitons maintain their shape but have effectively only one polarization component, while vector solitons have two distinct... from the perspect of experiments and numerical simulations. Different types of vector solitons and the polarization state of vector solitons have been investigated by his group. |

## Solitons in proteins and DNA

Solitons may occur in proteins and DNA. Solitons are related to the low-frequency collective motion in proteins and DNA.

## Solitons in magnets

In magnets, there also exist different types of solitons and other nonlinear waves. These magnetic solitons are an exact solution of classical nonlinear differential equations — magnetic equations, e.g. the

Landau–Lifshitz equationIn physics, the Landau–Lifshitz equation , named for Lev Landau and Evgeny Lifshitz, is a name used for several different differential equations*For the Landau–Lifshitz aeroacoustic equation see aeroacoustics....

, continuum

Heisenberg modelThe Heisenberg model can refer to two models in statistical mechanics:*Heisenberg model , a classical nearest neighbour spin model*Heisenberg model , a model where the spins are treated quantum mechanically using Pauli matrices....

,

Ishimori equationThe Ishimori equation is a partial differential equation proposed by the Japanese mathematician . Its interest is as the first example of a nonlinear spin-one field model in the plane that is integrable .-Equation:The IE has the form...

,

nonlinear Schrödinger equationIn theoretical physics, the nonlinear Schrödinger equation is a nonlinear version of Schrödinger's equation. It is a classical field equation with applications to optics and water waves. Unlike the Schrödinger equation, it never describes the time evolution of a quantum state...

and so on.

## 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 modelIn theoretical physics, the Born–Infeld model is a particular example of what is usually known as a nonlinear electrodynamics. It was historically introduced in the 30s to remove the divergence of the electron's self-energy in classical electrodynamics by introducing an upper bound of the electric...

. 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

- compacton, a soliton with compact support
- freak wave
Rogue waves are relatively large and spontaneous ocean surface waves that occur far out in sea, and are a threat even to large ships and ocean liners...

s may be a Peregrine solitonThe Peregrine soliton is an analytic solution of the nonlinear Schrödinger equation. This solution has been proposed in 1983 by Howell Peregrine, researcher at the mathematics department of the University of Bristol.- Main properties :...

related phenomenon involving BreatherA breather is a nonlinear wave in which energy concentrates in a localized and oscillatory fashion. This contradicts with the expectations derived from the corresponding linear system for infinitesimal amplitudes, which tends towards an even distribution of initially localized energy.A discrete...

waves which exhibit concentrated localized energy with non-linear properties.
- oscillon
In physics, an oscillon is a soliton-like phenomenon that occurs in granular and other dissipative media. Oscillons in granular media result from vertically vibrating a plate with a layer of uniform particles placed freely on top...

s
- peakon, a soliton with a non-differentiable peak
- soliton (topological)
- non-topological soliton
In quantum field theory, a non-topological soliton is a field configuration possessing, contrary to a topological one, a conserved Noether charge and stable against transformation into usual particles of this field for the following reason...

, in Quantum Field TheoryQuantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...

- Q-ball
In theoretical physics, Q-ball refers to a type of non-topological soliton. A soliton is a localized field configuration that is stable—it cannot spread out and dissipate. In the case of a non-topological soliton, the stability is guaranteed by a conserved charge: the soliton has lower energy per...

a non-topological soliton
- soliton (optics)
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 and linear effects in the medium. There are two main kinds of solitons:...

- soliton model
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 travel along the cell's membrane in the form of certain kinds of sound pulses known as solitons...

of nerve impulse propagation
- topological quantum number
In physics, a topological quantum number is any quantity, in a physical theory, that takes on only one of a discrete set of values, due to topological considerations...

- sine-Gordon 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 unknown function. It was originally considered in the nineteenth century in the course of study of surfaces of constant negative...

- nonlinear Schrödinger equation
In theoretical physics, the nonlinear Schrödinger equation is a nonlinear version of Schrödinger's equation. It is a classical field equation with applications to optics and water waves. Unlike the Schrödinger equation, it never describes the time evolution of a quantum state...

- vector soliton
In physical optics or wave optics, a vector soliton is a solitary wave with multiple components coupled together that maintains its shape during propagation. Ordinary solitons maintain their shape but have effectively only one polarization component, while vector solitons have two distinct...

- soliton distribution
A soliton distribution is a type of discrete probability distribution that arises in the theory of erasure correcting codes. A paper by Luby introduced two forms of such distributions, the ideal soliton distribution and the robust soliton distribution.-Ideal distribution:The ideal soliton...

- Soliton hypothesis for ball lightning, by David Finkelstein
David Ritz Finkelstein is currently an emeritus professor of physics at the Georgia Institute of Technology. Finkelstein obtained his Ph.D. in physics at the Massachusetts Institute of Technology in 1953. From 1964 to 1976, he was professor of physics at Yeshiva University.In 1958 Charles W...

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