# Soliton

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In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. (Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency.) Solitons are the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems.

## Contents

The soliton phenomenon was first described in 1834 by John Scott Russell (1808–1882) who observed a solitary wave in the Union Canal in Scotland. He reproduced the phenomenon in a wave tank and named it the "Wave of Translation".

## Definition

A single, consensus definition of a soliton is difficult to find. Drazin & Johnson (1989 , p. 15) ascribe three properties to solitons:

1. They are of permanent form;
2. They are localized within a region;
3. They can interact with other solitons, and emerge from the collision unchanged, except for a phase shift.

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 optics are often called solitons despite losing energy during interaction). [1]

## Explanation

Dispersion and nonlinearity 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 travel at different speeds and the shape of the pulse therefore changes over time. However, also the nonlinear Kerr effect occurs; the refractive index 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 exactly cancels the dispersion effect, and the pulse's shape does not change over time, thus is a soliton. See soliton (optics) for a more detailed description.

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, and owe their stability to the integrability of the field 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. 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 traveling 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.

A topological soliton, also called a topological defect, is any solution of a set of partial differential equations that is stable against decay to the "trivial solution". 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 nontrivial homotopy group, preserved by the differential equations. Thus, the differential equation solutions can be classified into homotopy classes.

No continuous transformation maps 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, the magnetic skyrmion in condensed matter physics, and cosmic strings and domain walls in cosmology.

## History

In 1834, John Scott Russell describes his wave of translation . [nb 1] The discovery is described here in Scott Russell's own words: [nb 2]

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. [2]

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 Newton's and Daniel Bernoulli's theories of hydrodynamics. George Biddell Airy and George Gabriel Stokes 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 [3] and Lord Rayleigh published a theoretical treatment and solutions. [nb 3] In 1895 Diederik Korteweg and Gustav de Vries provided what is now known as the Korteweg–de Vries equation, including solitary wave and periodic cnoidal wave solutions. [4] [nb 4]

In 1965 Norman Zabusky of Bell Labs and Martin Kruskal of Princeton University first demonstrated soliton behavior in media subject to the Korteweg–de Vries equation (KdV equation) in a computational investigation using a finite difference approach. They also showed how this behavior explained the puzzling earlier work of Fermi, Pasta, Ulam, and Tsingou. [6]

In 1967, Gardner, Greene, Kruskal and Miura discovered an inverse scattering transform enabling analytical solution of the KdV equation. [7] The work of Peter Lax on Lax pairs and the Lax equation has since extended this to solution of many related soliton-generating systems.

Note that solitons are, by definition, unaltered in shape and speed by a collision with other solitons. [8] So solitary waves on a water surface are near-solitons, but not exactly – after the interaction of two (colliding or overtaking) solitary waves, they have changed a bit in amplitude and an oscillatory residual is left behind. [9]

Solitons are also studied in quantum mechanics, thanks to the fact that they could provide a new foundation of it through de Broglie's unfinished program, known as "Double solution theory" or "Nonlinear wave mechanics". This theory, developed by de Broglie in 1927 and revived in the 1950s, is the natural continuation of his ideas developed between 1923 and 1926, which extended the wave-particle duality introduced by Albert Einstein for the light quanta, to all the particles of matter. In 2019, researchers from Tel-Aviv university measured an accelerating surface gravity water wave soliton by using an external hydrodynamic linear potential. They also managed to excite ballistic solitons and measure their corresponding phases. [10]

## 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 repeaters, and could potentially double transmission capacity as well. [11]

YearDiscovery
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. [12] Also in 1973 Robin Bullough 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 telecommunications.
1987 Emplit et al. (1987)  – from the Universities of Brussels and Limoges – made the first experimental observation of the propagation of a dark soliton, in an optical fiber.
1988Linn Mollenauer and his team transmitted soliton pulses over 4,000 kilometers using a phenomenon called the Raman effect, named after Sir C. V. Raman who first described it in the 1920s, to provide optical gain in the fiber.
1991A 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.
1998Thierry Georges and his team at France Telecom R&D Center, combining optical solitons of different wavelengths (wavelength-division multiplexing), demonstrated a composite data transmission of 1 terabit 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 dense wavelength-division multiplexing (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.

2000Cundiff predicted the existence of a vector soliton in a birefringence fiber cavity passively mode locking through a semiconductor saturable absorber mirror (SESAM). The polarization state of such a vector soliton could either be rotating or locked depending on the cavity parameters. [13]
2008D. Y. Tang et al. observed a novel form of higher-order vector soliton from the perspectives of experiments and numerical simulations. Different types of vector solitons and the polarization state of vector solitons have been investigated by his group. [14]

## In biology

Solitons may occur in proteins [15] and DNA. [16] Solitons are related to the low-frequency collective motion in proteins and DNA. [17]

A recently developed model in neuroscience proposes that signals, in the form of density waves, are conducted within neurons in the form of solitons. [18] [19] [20] Solitons can be described as almost lossless energy transfer in biomolecular chains or lattices as wave-like propagations of coupled conformational and electronic disturbances. [21]

## In magnets

In magnets, there also exist different types of solitons and other nonlinear waves. [22] These magnetic solitons are an exact solution of classical nonlinear differential equations — magnetic equations, e.g. the Landau–Lifshitz equation, continuum Heisenberg model, Ishimori equation, nonlinear Schrödinger equation and others.

## In nuclear physics

Atomic nuclei may exhibit solitonic behavior. [23] Here the whole nuclear wave function is predicted to exist as a soliton under certain conditions of temperature and energy. Such conditions are suggested to exist in the cores of some stars in which the nuclei would not react but pass through each other unchanged, retaining their soliton waves through a collision between nuclei.

The Skyrme Model is a model of nuclei in which each nucleus is considered to be a topologically stable soliton solution of a field theory with conserved baryon number.

## Bions

The bound state of two solitons is known as a bion, [24] [25] [26] or in systems where the bound state periodically oscillates, a breather .

In field theory bion usually refers to the solution of the Born–Infeld model. 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. [27] 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.

## Notes

1. "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 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 drift 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.
2. This passage has been repeated in many papers and books on soliton theory.
3. 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.
4. 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.

## Related Research Articles

In mathematics, the Korteweg–De Vries (KdV) 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 can be solved by means of the inverse scattering transform. The mathematical theory behind the KdV equation is a topic of active research. The KdV equation was first introduced by Boussinesq and rediscovered by Diederik Korteweg and Gustav de Vries (1895).

Martin 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. His most celebrated contribution was in the theory of solitons.

In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation. It is a classical field equation whose principal applications are to the propagation of light in nonlinear optical fibers and planar waveguides and to Bose–Einstein condensates confined to highly anisotropic cigar-shaped traps, in the mean-field regime. Additionally, the equation appears in the studies of small-amplitude gravity waves on the surface of deep inviscid (zero-viscosity) water; the Langmuir waves in hot plasmas; the propagation of plane-diffracted wave beams in the focusing regions of the ionosphere; the propagation of Davydov's alpha-helix solitons, which are responsible for energy transport along molecular chains; and many others. More generally, the NLSE appears as one of universal equations that describe the evolution of slowly varying packets of quasi-monochromatic waves in weakly nonlinear media that have dispersion. Unlike the linear Schrödinger equation, the NLSE never describes the time evolution of a quantum state. The 1D NLSE is an example of an integrable model.

In mathematics, and in particular in the theory of solitons, the Dym equation (HD) is the third-order partial differential equation

In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals, such that its behaviour has far fewer degrees of freedom than the dimensionality of its phase space; that is, its evolution is restricted to a submanifold within its phase space.

In mathematics and physics, the Kadomtsev–Petviashvili equation is a partial differential equation to describe nonlinear wave motion. Named after Boris Borisovich Kadomtsev and Vladimir Iosifovich Petviashvili, the KP equation is usually written as:

In 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. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to solve many linear partial differential equations. The name "inverse scattering method" comes from the key idea of recovering the time evolution of a potential from the time evolution of its scattering data: inverse scattering refers to the problem of recovering a potential from its scattering matrix, as opposed to the direct scattering problem of finding the scattering matrix from the potential.

The soliton hypothesis in neuroscience is a model that claims to explain how action potentials are initiated and conducted along axons based on a thermodynamic theory of nerve pulse propagation. It proposes that the signals travel along the cell's membrane in the form of certain kinds of solitary sound pulses that can be modeled as solitons. The model is proposed as an alternative to the Hodgkin–Huxley model in which action potentials: voltage-gated ion channels in the membrane open and allow sodium ions to enter the cell. The resulting decrease in membrane potential opens nearby voltage-gated sodium channels, thus propagating the action potential. The transmembrane potential is restored by delayed opening of potassium channels. Soliton hypothesis proponents assert that energy is mainly conserved during propagation except dissipation losses; Measured temperature changes are completely inconsistent with the Hodgkin-Huxley model.

Dissipative solitons (DSs) are stable solitary localized structures that arise in nonlinear spatially extended dissipative systems due to mechanisms of self-organization. They can be considered as an extension of the classical soliton concept in conservative systems. An alternative terminology includes autosolitons, spots and pulses.

In fluid dynamics, the Camassa–Holm equation is the integrable, dimensionless and non-linear partial differential equation

Norman J. Zabusky was 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 in the latter years of his career to an examination of the importance of visualization in this field. In fact, he coined the term visiometrics to describe the process of using computer-aided visualization to guide one towards quantitative results.

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 (scalar) polarization component, while vector solitons have two distinct polarization components. Among all the types of solitons, optical vector solitons draw the most attention due to their wide range of applications, particularly in generating ultrafast pulses and light control technology. Optical vector solitons can be classified into temporal vector solitons and spatial vector solitons. During the propagation of both temporal solitons and spatial solitons, despite being in a medium with birefringence, the orthogonal polarizations can copropagate as one unit without splitting due to the strong cross-phase modulation and coherent energy exchange between the two polarizations of the vector soliton which may induce intensity differences between these two polarizations. Thus vector solitons are no longer linearly polarized but rather elliptically polarized.

The Peregrine soliton is an analytic solution of the nonlinear Schrödinger equation. This solution was proposed in 1983 by Howell Peregrine, researcher at the mathematics department of the University of Bristol.

In theoretical physics, the logarithmic Schrödinger equation is one of the nonlinear modifications of Schrödinger's equation. It is a classical wave equation with applications to extensions of quantum mechanics, quantum optics, nuclear physics, transport and diffusion phenomena, open quantum systems and information theory, effective quantum gravity and physical vacuum models and theory of superfluidity and Bose–Einstein condensation. Its relativistic version was first proposed by Gerald Rosen. It is an example of an integrable model.

In mathematics, the Novikov–Veselov equation is a natural (2+1)-dimensional analogue of the Korteweg–de Vries (KdV) equation. Unlike another (2+1)-dimensional analogue of KdV, the Kadomtsev–Petviashvili equation, it is integrable via the inverse scattering transform for the 2-dimensional stationary Schrödinger equation. Similarly, the Korteweg–de Vries equation is integrable via the inverse scattering transform for the 1-dimensional Schrödinger equation. The equation is named after S.P. Novikov and A.P. Veselov who published it in Novikov & Veselov (1984).

Optical rogue waves are rare pulses of light analogous to rogue or freak ocean waves. The term optical rogue waves was coined to describe rare pulses of broadband light arising during the process of supercontinuum generation—a noise-sensitive nonlinear process in which extremely broadband radiation is generated from a narrowband input waveform—in nonlinear optical fiber. In this context, optical rogue waves are characterized by an anomalous surplus in energy at particular wavelengths and/or an unexpected peak power. These anomalous events have been shown to follow heavy-tailed statistics, also known as L-shaped statistics, fat-tailed statistics, or extreme-value statistics. These probability distributions are characterized by long tails: large outliers occur rarely, yet much more frequently than expected from Gaussian statistics and intuition. Such distributions also describe the probabilities of freak ocean waves and various phenomena in both the man-made and natural worlds. Despite their infrequency, rare events wield significant influence in many systems. Aside from the statistical similarities, light waves traveling in optical fibers are known to obey the similar mathematics as water waves traveling in the open ocean, supporting the analogy between oceanic rogue waves and their optical counterparts. More generally, research has exposed a number of different analogies between extreme events in optics and hydrodynamic systems. A key practical difference is that most optical experiments can be done with a table-top apparatus, offer a high degree of experimental control, and allow data to be acquired extremely rapidly. Consequently, optical rogue waves are attractive for experimental and theoretical research and have become a highly studied phenomenon. The particulars of the analogy between extreme waves in optics and hydrodynamics may vary depending on the context, but the existence of rare events and extreme statistics in wave-related phenomena are common ground.

The Kundu equation is a general form of integrable system that is gauge-equivalent to the mixed nonlinear Schrödinger equation. It was proposed by Anjan Kundu as

Bernstein–Greene–Kruskal modes are nonlinear electrostatic waves that propagate in an unmagnetized, collisionless plasma. They are nonlinear solutions to the Vlasov–Poisson equation in plasma physics, and are named after physicists Ira B. Bernstein, John M. Greene, and Martin D. Kruskal, who solved and published the exact solution for the one-dimensional case in 1957.

Integrable algorithms are numerical algorithms that rely on basic ideas from the mathematical theory of integrable systems.

The Schamel equation (S-equation) is a nonlinear partial differential equation of first order in time and third order in space. Similar to a Korteweg de Vries equation (KdV), it describes the development of a localized, coherent wave structure that propagates in a nonlinear dispersive medium. It was first derived in 1973 by Hans Schamel to describe the effects of electron trapping in the trough of the potential of a solitary electrostatic wave structure travelling with ion acoustic speed in a two-component plasma. It now applies to various localized pulse dynamics such as:

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