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104-588: 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 ". The term soliton was coined by Zabusky and Kruskal to describe localized, strongly stable propagating solutions to the Korteweg–de Vries equation , which models waves of

208-511: A functor on the category of topological spaces is homotopy invariant if it can be expressed as a functor on the homotopy category. For example, homology groups are a functorial homotopy invariant: this means that if f and g from X to Y are homotopic, then the group homomorphisms induced by f and g on the level of homology groups are the same: H n ( f ) = H n ( g ) : H n ( X ) → H n ( Y ) for all n . Likewise, if X and Y are in addition path connected , and

312-561: A homotopy equivalence between X and Y is a pair of continuous maps f  : X → Y and g  : Y → X , such that g  ∘  f is homotopic to the identity map id X and f  ∘  g is homotopic to id Y . If such a pair exists, then X and Y are said to be homotopy equivalent , or of the same homotopy type . Intuitively, two spaces X and Y are homotopy equivalent if they can be transformed into one another by bending, shrinking and expanding operations. Spaces that are homotopy-equivalent to

416-717: 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: 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

520-429: A classical paper and is quite frequently cited in soliton -related papers or books even after more than one hundred and fifty years. Homotopy class In topology , a branch of mathematics , two continuous functions from one topological space to another are called homotopic (from Ancient Greek : ὁμός homós "same, similar" and τόπος tópos "place") if one can be "continuously deformed" into

624-546: A collision with other solitons. 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. 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

728-472: A homotopy H  : X × [0,1] → Y and a cover p  : Y → Y and we are given a map h 0  : X → Y such that H 0 = p ○ h 0 ( h 0 is called a lift of h 0 ), then we can lift all H to a map H  : X × [0, 1] → Y such that p ○ H = H . The homotopy lifting property is used to characterize fibrations . Another useful property involving homotopy

832-736: A homotopy between two continuous functions f and g from a topological space X to a topological space Y is defined to be a continuous function H : X × [ 0 , 1 ] → Y {\displaystyle H:X\times [0,1]\to Y} from the product of the space X with the unit interval [0, 1] to Y such that H ( x , 0 ) = f ( x ) {\displaystyle H(x,0)=f(x)} and H ( x , 1 ) = g ( x ) {\displaystyle H(x,1)=g(x)} for all x ∈ X {\displaystyle x\in X} . If we think of

936-694: A house provided for the secretary of the Society of Arts and then moved to Sydenham Hill , which became a centre of attention especially after Russell and his friends moved Paxton 's glasshouse for the Great Exhibition to the Crystal Palace close by. Arthur Sullivan and his friend Frederic Clay were frequent visitors at the Scott Russell home in the mid-1860s; Clay became engaged to Alice, and Sullivan wooed Rachel. While Clay

1040-450: 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

1144-414: 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

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1248-657: A naval architect, contributing to the Institution of Naval Architects which his father had founded. While in Edinburgh he experimented with steam engines, using a square boiler for which he developed a method of staying the surface of the boiler which became universal. The Scottish Steam Carriage Company was formed producing a steam carriage with two cylinders developing 12 horsepower each. Six were constructed in 1834, well-sprung and fitted out to high standard, which from March 1834 ran between Glasgow's George Square and

1352-479: A path of embeddings: a continuous function starting at t  = 0 giving the K 1 embedding, ending at t  = 1 giving the K 2 embedding, with all intermediate values corresponding to embeddings. This corresponds to the definition of isotopy. An ambient isotopy , studied in this context, is an isotopy of the larger space, considered in light of its action on the embedded submanifold. Knots K 1 and K 2 are considered equivalent when there

1456-422: A point are called contractible . A homeomorphism is a special case of a homotopy equivalence, in which g  ∘  f is equal to the identity map id X (not only homotopic to it), and f  ∘  g is equal to id Y . Therefore, if X and Y are homeomorphic then they are homotopy-equivalent, but the opposite is not true. Some examples: A function f {\displaystyle f}

1560-599: A series of Royal Mail ships, together with many other innovations. After the shipyard was taken over by Caird , he decided to move to London and in 1848 purchased the Millwall Iron Works shipbuilding company. He built two ships for Brunel for the Australia run, much the same size as Brunel's SS Great Britain , Adelaide and Victoria . Problems with refuelling and water led Brunel to think in terms of larger ships for this voyage, but five more were built in

1664-407: A short time between London and Greenwich. In 1834, while conducting experiments to determine the most efficient design for canal boats, he discovered a phenomenon that he described as the wave of translation . In fluid dynamics the wave is now called Russell's solitary wave . The discovery is described here in his own words: I was observing the motion of a boat which was rapidly drawn along

1768-755: 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 . In 1834, John Scott Russell describes his wave of translation . The discovery

1872-402: 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

1976-584: A success that an international version was planned for 1851. By this time Russell had once again started shipbuilding, the railway boom having finished, and although he became the RSA's appointed secretary for the Great Exhibition , Henry Cole was by this time taking the lead, and he ended up with only a Gold Medal as his reward for much work. Russell became a member of the Institution of Civil Engineers in 1847 attending regularly and making frequent contributions,

2080-468: A time when all previous train ferries were riverine vessels, in 1868 Scott Russell designed a train ferry for service on Lake Constance , the Bodensee Trajekt , which entered service in 1869. This was the world's first cross-lake train ferry. The Bodensee Trajekt had to meet the unusual requirement that its draft not exceed six feet (1.85m). He achieved this by using the superstructure to carry

2184-574: A topologically stable soliton solution of a field theory with conserved baryon number. The bound state of two solitons is known as a bion , or in systems where the bound state periodically oscillates, a breather . The interference-type forces between solitons could be used in making bions. 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. The resulting self-generated potential profile features an inner attractive soft-core supporting

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2288-417: Is a subset of X , then we say that f and g are homotopic relative to K if there exists a homotopy H  : X × [0, 1] → Y between f and g such that H ( k ,  t ) = f ( k ) = g ( k ) for all k ∈ K and t ∈ [0, 1]. Also, if g is a retraction from X to K and f is the identity map, this is known as a strong deformation retract of X to K . When K

2392-475: 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

2496-399: Is a family of continuous functions h t : X → Y {\displaystyle h_{t}:X\to Y} for t ∈ [ 0 , 1 ] {\displaystyle t\in [0,1]} such that h 0 = f {\displaystyle h_{0}=f} and h 1 = g {\displaystyle h_{1}=g} , and

2600-536: Is a homotopy H taking f to g as described above. Being homotopic is an equivalence relation on the set of all continuous functions from X to Y . This homotopy relation is compatible with function composition in the following sense: if f 1 , g 1  : X → Y are homotopic, and f 2 , g 2  : Y → Z are homotopic, then their compositions f 2  ∘  f 1 and g 2  ∘  g 1  : X → Z are also homotopic. Given two topological spaces X and Y ,

2704-462: Is a homotopy such that the curve remains timelike during the continuous transformation from one curve to another. No closed timelike curve (CTC) on a Lorentzian manifold is timelike homotopic to a point (that is, null timelike homotopic); such a manifold is therefore said to be multiply connected by timelike curves. A manifold such as the 3-sphere can be simply connected (by any type of curve), and yet be timelike multiply connected . If we have

2808-447: Is a point, the term pointed homotopy is used. When two given continuous functions f and g from the topological space X to the topological space Y are embeddings , one can ask whether they can be connected 'through embeddings'. This gives rise to the concept of isotopy , which is a homotopy, H , in the notation used before, such that for each fixed t , H ( x ,  t ) gives an embedding. A related, but different, concept

2912-531: Is an ambient isotopy which moves K 1 to K 2 . This is the appropriate definition in the topological category. Similar language is used for the equivalent concept in contexts where one has a stronger notion of equivalence. For example, a path between two smooth embeddings is a smooth isotopy . On a Lorentzian manifold , certain curves are distinguished as timelike (representing something that only goes forwards, not backwards, in time, in every local frame). A timelike homotopy between two timelike curves

3016-515: Is denoted as "the law of Scott Russell" within the text. His contemporaries spent some time attempting to extend the theory but it would take until the 1870s before an explanation was provided. 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

3120-415: Is described here in Scott 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

3224-418: Is not homotopy-invariant is compactly supported homology (which is, roughly speaking, the homology of the compactification , and compactification is not homotopy-invariant). In order to define the fundamental group , one needs the notion of homotopy relative to a subspace . These are homotopies which keep the elements of the subspace fixed. Formally: if f and g are continuous maps from X to Y and K

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3328-444: Is not sufficient to require each map h t ( x ) {\displaystyle h_{t}(x)} to be continuous. The animation that is looped above right provides an example of a homotopy between two embeddings , f and g , of the torus into R . X is the torus, Y is R , f is some continuous function from the torus to R that takes the torus to the embedded surface-of-a-doughnut shape with which

3432-477: Is now known as the Korteweg–de Vries equation , including solitary wave and periodic cnoidal wave solutions. 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

3536-412: Is null-homotopic precisely when it can be continuously extended to a map from the unit disk D 2 {\displaystyle D^{2}} to X {\displaystyle X} that agrees with f {\displaystyle f} on the boundary. It follows from these definitions that a space X {\displaystyle X} is contractible if and only if

3640-411: Is said to be null-homotopic if it is homotopic to a constant function. (The homotopy from f {\displaystyle f} to a constant function is then sometimes called a null-homotopy .) For example, a map f {\displaystyle f} from the unit circle S 1 {\displaystyle S^{1}} to any space X {\displaystyle X}

3744-480: 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

3848-457: Is that of ambient isotopy . Requiring that two embeddings be isotopic is a stronger requirement than that they be homotopic. For example, the map from the interval [−1, 1] into the real numbers defined by f ( x ) = − x is not isotopic to the identity g ( x ) = x . Any homotopy from f to the identity would have to exchange the endpoints, which would mean that they would have to 'pass through' each other. Moreover, f has changed

3952-426: Is the homotopy extension property , which characterizes the extension of a homotopy between two functions from a subset of some set to the set itself. It is useful when dealing with cofibrations . Since the relation of two functions f , g : X → Y {\displaystyle f,g\colon X\to Y} being homotopic relative to a subspace is an equivalence relation, we can look at

4056-517: 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. The word regular means a smooth solution carrying no sources at all. However, the solution of the Born–Infeld model still carries

4160-496: 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

4264-457: 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

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4368-404: The equivalence classes of maps between a fixed X and Y . If we fix X = [ 0 , 1 ] n {\displaystyle X=[0,1]^{n}} , the unit interval [0, 1] crossed with itself n times, and we take its boundary ∂ ( [ 0 , 1 ] n ) {\displaystyle \partial ([0,1]^{n})} as a subspace, then

4472-470: The map ( x , t ) ↦ h t ( x ) {\displaystyle (x,t)\mapsto h_{t}(x)} is continuous from X × [ 0 , 1 ] {\displaystyle X\times [0,1]} to Y {\displaystyle Y} . The two versions coincide by setting h t ( x ) = H ( x , t ) {\displaystyle h_{t}(x)=H(x,t)} . It

4576-424: 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. 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

4680-414: 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 theory bion usually refers to the solution of

4784-592: The Edinburgh Journal of Science in the same year. 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 contrast with Isaac Newton 's and Daniel Bernoulli 's theories of hydrodynamics . George Biddell Airy and George Gabriel Stokes had difficulty to accept Scott Russell's experimental observations because Scott Russell's observations could not be explained by

4888-625: The French Academy of Sciences on the portion of Bazin's treatise relating to surges and the propagation of waves ) was featured in the proceedings of the Dutch Koninklijk Instituut van Ingenieurs (English: Royal Institute of Engineers ) in 1869. Within the original French paper, and the translated work, the velocity of a solitary wave is given as: c = g ( h + z ) {\displaystyle c={\sqrt {g(h+z)}}} The formula

4992-707: The Gordon–Mollenauer effect. Consequently, the long-haul fiberoptic transmission soliton has remained a laboratory curiosity. Solitons may occur in proteins and DNA. Solitons are related to the low-frequency collective motion in proteins and DNA . A recently developed model in neuroscience proposes that signals, in the form of density waves, are conducted within neurons in the form of solitons. Solitons can be described as almost lossless energy transfer in biomolecular chains or lattices as wave-like propagations of coupled conformational and electronic disturbances. Solitons can occur in materials, such as ferroelectrics , in

5096-883: The Laws by which water opposes Resistance to the Motion of Floating Bodies". He was elected a Fellow of the Royal Society in June 1849 for Memoirs on "The great Solitary Wave of the First Order, or the Wave of Translation" published in the Transactions of the Royal Society of Edinburgh, and of several Memoirs in the Reports of the British Association . In 1995, the aqueduct which carries the Union Canal –

5200-562: The Leith Mechanics' Institute , achieving the highest attendance in the city. On the death of Sir John Leslie , Professor of Natural Philosophy at the University of Edinburgh in 1832, Scott Russell, though only 24 years old, was elected to temporarily fill the vacancy pending the election of a permanent professor, due to his proficiency in the natural sciences and popularity as a lecturer. But although encouraged to stand for

5304-592: The Tontine Hotel in Paisley at hourly intervals at 15 mph. The road trustees objected that it wore out the road and placed various obstructions of logs and stones in the road, which actually caused more discomfort for horse-drawn carriages. But in July 1834 one of the carriages was overturned and the boiler smashed, causing the death of several passengers. Two of the coaches were sent to London where they ran for

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5408-584: The action of one equivalence class on another, and so we get a group. These groups are called the homotopy groups . In the case n = 1 {\displaystyle n=1} , it is also called the fundamental group . The idea of homotopy can be turned into a formal category of category theory . The homotopy category is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. Two topological spaces X and Y are isomorphic in this category if and only if they are homotopy-equivalent. Then

5512-461: The animation starts; g is some continuous function that takes the torus to the embedded surface-of-a-coffee-mug shape. The animation shows the image of h t (X) as a function of the parameter t , where t varies with time from 0 to 1 over each cycle of the animation loop. It pauses, then shows the image as t varies back from 1 to 0, pauses, and repeats this cycle. Continuous functions f and g are said to be homotopic if and only if there

5616-455: The cargo-carrying capacity, but starting from the 1840s the "extreme clipper ships " started to show concave bows as increasingly did steam ships culminating with Great Eastern . After his views were propounded by Commander Fishbourne, the American naval architect John W. Griffiths acknowledged the force of Russell's work in his Treatise on marine and naval architecture of 1850 though he

5720-413: 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 [14 km/h], preserving its original figure some thirty feet [9 m] long and a foot to a foot and a half [30−45 cm] in height. Its height gradually diminished, and after a chase of one or two miles [2–3 km] I lost it in

5824-409: The continuation method (see numerical continuation ). The methods for differential equations include the homotopy analysis method . Homotopy theory can be used as a foundation for homology theory : one can represent a cohomology functor on a space X by mappings of X into an appropriate fixed space, up to homotopy equivalence. For example, for any abelian group G , and any based CW-complex X ,

5928-560: The coupling of the Born–Infeld model to general relativity) the corresponding solution is called EBIon , where "E" stands for Einstein. Erik Lentz, a physicist at the University of Göttingen, has theorized that solitons could allow for the generation of Alcubierre warp bubbles in spacetime without the need for exotic matter, i.e., matter with negative mass. John Russell (engineer) John Scott Russell (9 May 1808, Parkhead , Glasgow – 8 June 1882, Ventnor , Isle of Wight)

6032-620: The domains, influencing the direction of the soliton network propagation. 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. In addition, closed loops of domain walls enclosing multiple polarization domains can inhibit soliton propagation and thus, switching of polarizations across it. Also, domain walls can propagate and meet at wrinkles and surface inhomogeneities within

6136-406: The equivalence classes form a group, denoted π n ( Y , y 0 ) {\displaystyle \pi _{n}(Y,y_{0})} , where y 0 {\displaystyle y_{0}} is in the image of the subspace ∂ ( [ 0 , 1 ] n ) {\displaystyle \partial ([0,1]^{n})} . We can define

6240-481: The existing water wave theories. Additional observations were reported by Henry Bazin in 1862 after experiments carried out in the canal de Bourgogne in France. 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. In 1895 Diederik Korteweg and Gustav de Vries provided what

6344-608: The existing water-wave theories. Additional observations were reported by Henry Bazin in 1862 after experiments carried out in the canal de Bourgogne in France. In 1863, Bazin authored a research paper titled Recherches hydrauliques entreprises par M.H. Darcy (English: Hydraulic Researches Undertaken by M.H. Darcy ) which featured the work of Scott Russell. A Dutch translation of Bazin's work entitled Verslag aan de Fransche academie van wetenschappen over het gedeelte der verhandeling van Bazin, betrekkelijk de opstuwingen en de voortbeweging der golven (English: Report to

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6448-517: The first theoretical treatment was by Joseph Valentin Boussinesq in 1871; Boussinesq had mentioned Scott Russell's name in his 1871 paper. Thus Scott Russell's observations on solitary waves were accepted as true by some prominent scientists within his own lifetime. 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. Although

6552-439: 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

6656-541: 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 of lattice dislocations . The domain walls can propagate as

6760-404: The front of his wave of translation, he tackled the more fundamental issue for boat design of finding the hull shape which gives the least resistance. This, he reasoned was concerned with moving the mass of water efficiently out of the way of the hull and then back to fill the gap after it has passed. By careful measurements with dynamometers he validated his theory that a versed sine wave produces

6864-429: The homotopy between f and g is pointed, then the group homomorphisms induced by f and g on the level of homotopy groups are also the same: π n ( f ) = π n ( g ) : π n ( X ) → π n ( Y ). Based on the concept of the homotopy, computation methods for algebraic and differential equations have been developed. The methods for algebraic equations include the homotopy continuation method and

6968-404: The ideal shape. Initially he thought that the stern could be a mirror of the stem , but soon realised that the removing water produced something closer to conventional waves than his solitary waves and ended up with a rounded stern with a catenary shape. His studies produced a revolution in the design of hulls for merchant and navy vessels. Most ships of the time had rounded bows to optimise

7072-449: The identity map from X {\displaystyle X} to itself—which is always a homotopy equivalence—is null-homotopic. Homotopy equivalence is important because in algebraic topology many concepts are homotopy invariant , that is, they respect the relation of homotopy equivalence. For example, if X and Y are homotopy equivalent spaces, then: An example of an algebraic invariant of topological spaces which

7176-421: 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 an AFM 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. It has also been observed that

7280-424: The map of the unit disc in R defined by f ( x ,  y ) = (− x , − y ) is isotopic to a 180-degree rotation around the origin, and so the identity map and f are isotopic because they can be connected by rotations. In geometric topology —for example in knot theory —the idea of isotopy is used to construct equivalence relations. For example, when should two knots be considered

7384-440: The modern general theory of solitons. Note that solitons are, by definition, unaltered in shape and speed by a collision with other solitons. So solitary waves on a water surface are not solitons – after the interaction of two (colliding or overtaking) solitary waves, they have changed slightly in amplitude and an oscillatory residual is left behind. Once Russell had a way of observing boats at hitherto unprecedented speeds at

7488-482: The orientation of the interval and g has not, which is impossible under an isotopy. However, the maps are homotopic; one homotopy from f to the identity is H : [−1, 1] × [0, 1] → [−1, 1] given by H ( x ,  y ) = 2 yx  −  x . Two homeomorphisms (which are special cases of embeddings) of the unit ball which agree on the boundary can be shown to be isotopic using Alexander's trick . For this reason,

7592-546: The other, such a deformation being called a homotopy ( / h ə ˈ m ɒ t ə p iː / , hə- MO -tə-pee ; / ˈ h oʊ m oʊ ˌ t oʊ p iː / , HOH -moh-toh-pee ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups , important invariants in algebraic topology . In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces , CW complexes , or spectra . Formally,

7696-414: The paper by Korteweg and de Vries in 1895 was not the first theoretical treatment of this subject, it was a very important milestone in the history of the development of soliton theory. It was not until the 1960s and the advent of modern computers that the significance of Scott Russell's discovery in physics , electronics , biology and especially fibre optics started to become understood, leading to

7800-646: The permanent position he refused to compete with another candidate he admired and thereafter concentrated the engineering profession and experimental research on a large scale. He married Harriette Osborne, daughter of the Irish baronet Sir Daniel Toler Osborne and Harriette Trench, daughter of the Earl of Clancarty in Dublin in 1839; they had two sons (Norman survived) and three daughters, Louise (1841–1878), Rachel (1845–1882) and Alice. In London they lived for five years in

7904-578: The phases of ballistic solitons. 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. The above impressive experiments have not translated to actual commercial soliton system deployments however, in either terrestrial or submarine systems, chiefly due to

8008-535: 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.   In recent literature, ferroelectricity has been observed in twisted bilayers of van der Waal materials such as molybdenum disulfide and graphene . The moiré superlattice that arises from

8112-507: The puzzling earlier work of Fermi, Pasta, Ulam, and Tsingou . In 1967, Gardner, Greene, Kruskal and Miura discovered an inverse scattering transform enabling analytical solution of the KdV equation. The work of Peter Lax on Lax pairs and the Lax equation has since extended this to solution of many related soliton-generating systems. Solitons are, by definition, unaltered in shape and speed by

8216-711: The railway boom was at its height. Russell had contributed an article on the Steam engine and steam navigation for the 7th edition of Encyclopædia Britannica in 1841 which also appeared in book form. Charles Wentworth Dilke offered him the editorial position of a new weekly paper, the Railway Chronicle in London and the Russell family was soon in a small two-room flat in Westminster . The next year he also became

8320-438: 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 of partial dislocations where different types of stresses, and thus, strains are experienced by

8424-666: The same canal where he observed his Wave of Translation – over the Edinburgh Bypass (A720) was named the Scott Russell Aqueduct in his memory. Also in 1995, the hydrodynamic soliton effect was reproduced near the place where John Scott Russell observed hydrodynamic solitons in 1834. A building at Heriot-Watt University is named after him. In 2019 he was inducted into the Scottish Engineering Hall of Fame His 1844 paper has become

8528-506: The same class. Before they started any business together, he was held in high regard by Isambard Kingdom Brunel who made him a partner in his project to build Great Eastern . Although the original conception, the cellular construction and the joint use of paddle and screw were Brunel's ideas, "the ship embodies the wave-line form, the longitudinal system of construction, the complete and partial bulkheads, and other details of construction which were peculiarly Scott Russell’s". The project

8632-403: The same? We take two knots, K 1 and K 2 , in three- dimensional space. A knot is an embedding of a one-dimensional space, the "loop of string" (or the circle), into this space, and this embedding gives a homeomorphism between the circle and its image in the embedding space. The intuitive idea behind the notion of knot equivalence is that one can deform one embedding to another through

8736-512: The second parameter of H as time then H describes a continuous deformation of f into g : at time 0 we have the function f and at time 1 we have the function g . We can also think of the second parameter as a "slider control" that allows us to smoothly transition from f to g as the slider moves from 0 to 1, and vice versa. An alternative notation is to say that a homotopy between two continuous functions f , g : X → Y {\displaystyle f,g:X\to Y}

8840-563: The secretary of the committee set up by the Royal Society of Arts to organise a national exhibition, which provided them with a town house in the Strand . Russell soon introduced Henry Cole to the committee and when, a few weeks before the first exhibition in 1847, there were no exhibitors, Russell and Cole spent three whole days travelling around London to enlist manufacturers and shopkeepers. This and two subsequent exhibitions were such

8944-439: The set [ X , K ( G , n ) ] {\displaystyle [X,K(G,n)]} of based homotopy classes of based maps from X to the  Eilenberg–MacLane space K ( G , n ) {\displaystyle K(G,n)} is in natural bijection with the n -th singular cohomology group H n ( X , G ) {\displaystyle H^{n}(X,G)}  of

9048-455: 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, the pulse is a soliton. See soliton (optics) for

9152-422: The son of Reverend David Russell and Agnes Clark Scott. He spent one year at the University of St. Andrews before transferring to the University of Glasgow . It was while at the University of Glasgow that he added his mother's maiden name, Scott, to his own, to become John Scott Russell. He graduated from Glasgow University in 1825 at the age of 17 and moved to Edinburgh where he taught mathematics and science at

9256-533: The stresses of the train. (It was not until 1892 that the first Lake Michigan cross-lake train ferry, the Ann Arbor No. 1 , designed by Frank E. Kirby , entered service.) Scott Russell used the design of the Bodensee Trajekt as the basis of a cross-channel ferry that could manage the shallow harbour of Dover, but this was not realised until 1933. Although his design for the Great Exhibition

9360-525: 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). 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

9464-591: The time of the construction of Great Eastern . Holley also visited Scott Russell's house in Sydenham. As a result of this, Holley, and his colleague Zerah Colburn , travelled on the maiden voyage of Great Eastern from Southampton to New York in June 1860. Scott Russell's son, Norman, stayed with Holley at his house in Brooklyn — Norman also travelled on the maiden voyage, one voyage that John Scott Russell did not make. His son, Norman, followed his father in becoming

9568-402: 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. Different slip directions of the walls are achieved with different types of strains found at

9672-500: The type seen by Russell. The name was meant to characterize the solitary nature of the waves, with the 'on' suffix recalling the usage for particles such as electrons , baryons or hadrons , reflecting their observed particle-like behaviour. 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

9776-529: The van der Waal layers, which can act as obstacles obstructing the propagation. 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 equation , continuum Heisenberg model , Ishimori equation , nonlinear Schrödinger equation and others. Atomic nuclei may exhibit solitonic behavior. Here

9880-410: 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

9984-520: 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. The phenomenon of solitary waves had previously been reported in 1826 by Giorgio Bidone in Turin , but Bidone's work seems to have gone unnoticed by researchers in the Netherlands and Britain, despite being mentioned in

10088-487: Was a Scottish civil engineer , naval architect and shipbuilder who built Great Eastern in collaboration with Isambard Kingdom Brunel . He made the discovery of the wave of translation that gave birth to the modern study of solitons , and developed the wave-line system of ship construction. Russell was a promoter of the Great Exhibition of 1851 . John Russell was born on the 9th May 1808 with in Parkhead, Glasgow,

10192-435: Was elected to its council in 1857 and became a vice-president in 1862. However he became involved in a financial dispute with Sir William Armstrong and didn't become president. But "as a speaker, and particularly as an after-dinner speaker, he had few equals." He was elected a Fellow of the Royal Society in 1849 although he contributed less. In 1860 at a meeting at his house in Sydenham, the Institution of Naval Architects

10296-622: Was from a wealthy family, Sullivan was still a poor young composer from a poor family; the Scott Russells welcomed the engagement of Alice to Clay, who, however broke it off, but forbade the relationship between Sullivan and Rachel, although the two continued to see each other covertly. At some point in 1868, Sullivan started a simultaneous (and secret) affair with Louise (1841–1878). Both relationships had ceased by early 1869. The American engineer Alexander Lyman Holley befriended Scott Russell and his family on his various visits to London at

10400-594: Was grudging in acknowledging a debt to Russell. Scott Russell made one of the first experimental observations of the Doppler effect which he published in 1848. Christian Doppler published his theory in 1842. Much of Russell's early experimental work had been conducted under the auspices of the British Association and throughout his life he contributed to the scientific and professional associations that were becoming more important in that era. In 1844,

10504-479: Was now becoming the more lamentably apparent with every day that passed". During the 1850s he argued within the Navy for the construction of iron warships and the first design, HMS  Warrior , is said by some to be a "Russell ship". He afterwards complained about the secrecy that prevented an open discussion of the issues, criticizing those within the Navy who argued that iron ships could not be protected. At

10608-614: Was plagued with a number of problems—Scott Russell put in a bid which was far too low with the result that he was bankrupt halfway through, though he recovered to finish the job; but it was Brunel that insisted on a sideways launch rather than the dry dock that Russell preferred. Great Eastern was eventually launched in 1858. Scott Russell was a better scientist than a businessman and his reputation never fully recovered from his financial irregularities, gross neglect of duty and disputes. As L. T. C. Rolt writes in his biography of Brunel "That Russell had indeed misled Brunel and betrayed his trust

10712-475: Was set up, with Russell as one of the professional vice-presidents. He attended most meetings and rarely failed to comment. In 1864 he published a massive 3-volume treatise on The Modern System of Naval Architecture which laid out the profiles of many of the new ships being built. His obituary said of naval architecture: From around 1838, Scott Russell was employed at the small Greenock shipyard of Thomson and Spiers where he introduced his wave-line system to

10816-471: Was trumped by that of Joseph Paxton , Scott Russell did design the Rotunde for the 1873 Vienna Exposition . At 108 metres (354 ft) in diameter it was for nearly a century the largest cupola in the world, having no ties to obstruct the view. Some consider it his greatest structural engineering achievement. In 1838 he was awarded the gold Keith Medal by the Royal Society of Edinburgh for his paper "On

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