Inmathematics andphysics, asoliton is a nonlinear, self-reinforcing, localizedwave packet that isstrongly stable, in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such localized wave packets. Its remarkable stability can be traced to a balanced cancellation ofnonlinear anddispersive effects in the medium.[nb 1] Solitons were subsequently found to provide stable solutions of a wide class of weakly nonlinear dispersivepartial differential equations describing physical systems.
The soliton phenomenon was first described in 1834 byJohn Scott Russell who observed a solitary wave in theUnion Canal in Scotland. He reproduced the phenomenon in awave tank and named it the "Wave of Translation". TheKorteweg–de Vries equation was later formulated to model such waves, and the term "soliton" was coined byNorman Zabusky andMartin David Kruskal to describe localized, strongly stable propagating solutions to this equation. The name was meant to characterize the solitary nature of the waves, with the "on" suffix recalling the usage for particles such aselectrons,baryons orhadrons, reflecting their observedparticle-like behaviour.[1]
A single, consensus definition of a soliton is difficult to find.Drazin & Johnson (1989, p. 15) ascribe three properties to solitons:
More formal definitions exist, but they require substantial mathematics. Moreover, some scientists use the termsoliton for phenomena that do not quite have these three properties (for instance, the 'light bullets' ofnonlinear optics are often called solitons despite losing energy during interaction).[2]
Dispersion andnonlinearity can interact to produce permanent and localizedwave 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 nonlinearKerr effect occurs; therefractive 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, the pulse is a soliton. Seesoliton (optics) for a more detailed description.
Manyexactly solvable models have soliton solutions, including theKorteweg–de Vries equation, thenonlinear Schrödinger equation, the coupled nonlinear Schrödinger equation, and thesine-Gordon equation. The soliton solutions are typically obtained by means of theinverse scattering transform, and owe their stability to theintegrability of the field equations. The mathematical theory of these equations is a broad and very active field of mathematical research.
Some types oftidal bore, a wave phenomenon of a few rivers including theRiver Severn, are 'undular': a wavefront followed by a train of solitons. Other solitons occur as the underseainternal waves, initiated byseabed topography, that propagate on the oceanicpycnocline. Atmospheric solitons also exist, such as themorning glory cloud of theGulf of Carpentaria, where pressure solitons traveling in atemperature inversion layer produce vast linearroll clouds. The recent and not widely acceptedsoliton model inneuroscience proposes to explain the signal conduction withinneurons as pressure solitons.
Atopological soliton, also called a topological defect, is any solution of a set ofpartial 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 ofboundary conditions, and the boundary has a nontrivialhomotopy group, preserved by the differential equations. Thus, the differential equation solutions can be classified intohomotopy classes.
No continuous transformation maps a soliton in one homotopy class to another. The solitons are truly distinct, and maintain their integrity, even in the face of extremely powerful forces. Examples of topological solitons include thescrew dislocation in acrystalline lattice, theDirac string and themagnetic monopole inelectromagnetism, theSkyrmion and theWess–Zumino–Witten model inquantum field theory, themagnetic skyrmion in condensed matter physics, andcosmic strings anddomain walls incosmology.
In 1834,John Scott Russell described hiswave of translation:[nb 2][nb 3]
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.[3]
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:
Scott Russell's experimental work seemed at odds withIsaac Newton's andDaniel Bernoulli's theories ofhydrodynamics.George Biddell Airy andGeorge Gabriel Stokes had difficulty accepting Scott Russell's experimental observations because they could not be explained by the existing water wave theories. Additional observations were reported byHenry Bazin in 1862 after experiments carried out in thecanal de Bourgogne in France.[4] Their contemporaries spent some time attempting to extend the theory but it would take until the 1870s beforeJoseph Boussinesq[5] andLord Rayleigh published a theoretical treatment and solutions.[nb 4] In 1895Diederik Korteweg andGustav de Vries provided what is now known as theKorteweg–de Vries equation, including solitary wave and periodiccnoidal wave solutions.[6][nb 5]
In 1965Norman Zabusky ofBell Labs andMartin Kruskal ofPrinceton University first demonstrated soliton behavior in media subject to theKorteweg–de Vries equation (KdV equation) in a computational investigation using afinite difference approach. They also showed how this behavior explained the puzzling earlier work ofFermi, Pasta, Ulam, and Tsingou.[1]
In 1967, Gardner, Greene, Kruskal and Miura discovered aninverse scattering transform enablinganalytical solution of the KdV equation.[8] The work ofPeter Lax onLax pairs and the Lax equation has since extended this to solution of many related soliton-generating systems.
Solitons are, by definition, unaltered in shape and speed by a collision with other solitons.[9] So solitary waves on a water surface arenear-solitons, but not exactly – after the interaction of two (colliding or overtaking) solitary waves, they have changed a bit inamplitude and an oscillatory residual is left behind.[10]
Solitons are also studied in quantum mechanics, thanks to the fact that they could provide a new foundation of it throughde 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 thewave–particle duality introduced byAlbert Einstein for thelight quanta, to all the particles of matter. The observation of accelerating surface gravity water wave soliton using an external hydrodynamic linear potential was demonstrated in 2019. This experiment also demonstrated the ability to excite and measure the phases of ballistic solitons.[11]
Much experimentation has been done using solitons in fiber optics applications. Solitons in a fiber optic system are described by theManakov equations.Solitons' inherent stability make long-distance transmission possible without the use ofrepeaters, and could potentially double transmission capacity as well.[12]
| Year | Discovery |
|---|---|
| 1973 | Akira Hasegawa ofAT&TBell Labs was the first to suggest that solitons could exist inoptical fibers, due to a balance betweenself-phase modulation andanomalous dispersion.[13] Also in 1973Robin 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 opticaltelecommunications. |
| 1987 | Emplit et al. (1987) – from the Universities of Brussels and Limoges – made the first experimental observation of the propagation of adark soliton, in an optical fiber. |
| 1988 | Linn F. Mollenauer and his team transmitted soliton pulses over 4,000 kilometers using a phenomenon called theRaman effect, named afterSir C. V. Raman who first described it in the 1920s, to provideoptical 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, usingerbium 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 atFrance Telecom R&D Center, combining optical solitons of differentwavelengths (wavelength-division multiplexing), demonstrated acomposite data transmission of 1terabit 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 theGordon–Haus (GH) jitter. The GH jitter requires sophisticated, expensive compensatory solutions that ultimately makesdense 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. |
| 2000 | Steven Cundiff predicted the existence of avector soliton in a birefringence fiber cavity passively mode locking through asemiconductor saturable absorber mirror (SESAM). The polarization state of such a vector soliton could either be rotating or locked depending on the cavity parameters.[14] |
| 2008 | D. Y. Tanget al. observed a novel form ofhigher-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.[15] |
Solitons may occur in proteins[16] and DNA.[17] Solitons are related to thelow-frequency collective motion in proteins and DNA.[18]
A recently developedmodel in neuroscience proposes that signals, in the form of density waves, are conducted within neurons in the form of solitons.[19][20][21] Solitons can be described as almost lossless energy transfer in biomolecular chains or lattices as wave-like propagations of coupled conformational and electronic disturbances.[22]
Solitons can occur in materials, such asferroelectrics, in the form of domain walls. Ferroelectric materials exhibit spontaneous polarization, or electric dipoles, which are coupled to configurations of the material structure. Domains of oppositely poled polarizations can be present within a single material as the structural configurations corresponding to opposing polarizations are equally favorable with no presence of external forces. The domain boundaries, or "walls", that separate these local structural configurations are regions oflattice dislocations.[23] The domain walls can propagate as the polarizations, and thus, the local structural configurations can switch within a domain with applied forces such as electric bias or mechanical stress. Consequently, the domain walls can be described as solitons, discrete regions of dislocations that are able to slip or propagate and maintain their shape in width and length.[24][25][26]
In recent literature, ferroelectricity has been observed in twisted bilayers ofvan der Waal materials such asmolybdenum disulfide andgraphene.[23][27][28] Themoirésuperlattice that arises from the relative twist angle between the van der Waal monolayers generates regions of different stacking orders of the atoms within the layers. These regions exhibit inversion symmetry breaking structural configurations that enable ferroelectricity at the interface of these monolayers. The domain walls that separate these regions are composed ofpartial dislocations where different types of stresses, and thus, strains are experienced by the lattice. It has been observed that soliton or domain wall propagation across a moderate length of the sample (order of nanometers to micrometers) can be initiated with applied stress from anAFM tip on a fixed region. The soliton propagation carries the mechanical perturbation with little loss in energy across the material, which enables domain switching in a domino-like fashion.[25]
It has also been observed that the type of dislocations found at the walls can affect propagation parameters such as direction. For instance,STM measurements showed four types of strains of varying degrees of shear, compression, and tension at domain walls depending on the type of localized stacking order in twisted bilayer graphene. Differentslip directions of the walls are achieved with different types of strains found at the domains, influencing the direction of the soliton network propagation.[25]
Nonidealities such as disruptions to the soliton network and surface impurities can influence soliton propagation as well. Domain walls can meet at nodes and get effectively pinned, forming triangular domains, which have been readily observed in various ferroelectric twisted bilayer systems.[23] In addition, closed loops of domain walls enclosing multiple polarization domains can inhibit soliton propagation and thus, switching of polarizations across it.[25] Also, domain walls can propagate and meet at wrinkles and surface inhomogeneities within the van der Waal layers, which can act as obstacles obstructing the propagation.[25]
In magnets, there also exist different types of solitons and other nonlinear waves.[29] These magnetic solitons are an exact solution of classical nonlinear differential equations — magnetic equations, e.g. theLandau–Lifshitz equation, continuumHeisenberg model,Ishimori equation,nonlinear Schrödinger equation and others.
Atomic nuclei may exhibit solitonic behavior.[30] 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.
TheSkyrme 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.
The bound state of two solitons is known as abion,[31][32][33][34] or in systems where the bound state periodically oscillates, abreather. The interference-type forces between solitons could be used in making bions.[35] However, these forces are very sensitive to their relative phases. Alternatively, the bound state of solitons could be formed by dressing atoms with highly excited Rydberg levels.[34] The resulting self-generated potential profile[34] features an inner attractive soft-core supporting the 3D self-trapped soliton, an intermediate repulsive shell (barrier) preventing solitons' fusion, and an outer attractive layer (well) used for completing the bound state resulting in giant stable soliton molecules. In this scheme, the distance and size of the individual solitons in the molecule can be controlled dynamically with the laser adjustment.
In field theorybion usually refers to the solution of theBorn–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 aregular, finite-energy (and usually stable) solution of a differential equation describing some physical system.[36] The wordregular 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 calledEBIon, where "E" stands for Einstein.
In work that has been shown to violate known physics,[37]solitons have been proposed that could allow for the generation ofAlcubierre warp bubbles in spacetime without the need for exotic matter, i.e., matter with negative mass.[38][39][40]
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