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Soliton
In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is , 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 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 were subsequently found to provide stable solutions of a wide class of weakly nonlinear dispersive partial differential equations describing physical systems. The soliton phenomenon was first described in 1834 by John Scott Russell 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 Korteweg–de Vries equation was later formulated to model such waves, and the term "soliton" was coined by Zabu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Soliton Hydro
In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is , 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 of nonlinearity, nonlinear and dispersion relation, dispersive effects in the medium.Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency. Solitons were subsequently found to provide stable solutions of a wide class of weakly nonlinear dispersive partial differential equations describing physical systems. The soliton phenomenon was first described in 1834 by John Russell (engineer), John Scott Russell who observed a solitary wave in the Union Canal (Scotland), Union Canal in Scotland. He reproduced the phenomenon in a wave tank and named it the "John Russell (engineer)#The wave of translation, Wave of Trans ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 dispersive effects in the medium. There are two main kinds of solitons: * spatial solitons: the nonlinear effect can balance the dispersion. The electromagnetic field can change the refractive index of the medium while propagating, thus creating a structure similar to a graded-index fiber. If the field is also a propagating mode of the guide it has created, then it will remain confined and it will propagate without changing its shape * temporal solitons: if the electromagnetic field is already spatially confined, it is possible to send pulses that will not change their shape because the nonlinear effects will balance the dispersion. Those solitons were discovered first and they are often simply referred as "solitons" in optics. *Spatial solitons* In order to understand how a spatial soliton can exist, we have to make som ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sine-Gordon Equation
The sine-Gordon equation is a second-order nonlinear partial differential equation for a function \varphi dependent on two variables typically denoted x and t, involving the wave operator and the sine of \varphi. It was originally introduced by in the course of study of surfaces of constant negative curvature as the Gauss–Codazzi equation for surfaces of constant Gaussian curvature −1 in 3-dimensional space. The equation was rediscovered by in their study of crystal dislocations known as the Frenkel–Kontorova model. This equation attracted a lot of attention in the 1970s due to the presence of soliton solutions, and is an example of an integrable PDE. Among well-known integrable PDEs, the sine-Gordon equation is the only ''relativistic'' system due to its Lorentz invariance. Realizations of the sine-Gordon equation Differential geometry This is the first derivation of the equation, by Bour (1862). There are two equivalent forms of the sine-Gordon equation. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Scattering Transform
In mathematics, the inverse scattering transform is a method that solves the initial value problem for a Nonlinear system, nonlinear partial differential equation using mathematical methods related to scattering, wave scattering. The direct scattering transform describes how a Function (mathematics), function scatters waves or generates Bound state, bound-states. The inverse scattering transform uses wave scattering data to construct the function responsible for wave scattering. The direct and inverse scattering transforms are analogous to the direct and inverse Fourier transforms which are used to solve Linear differential equation, linear partial differential equations. Using a pair of differential operators, a 3-step algorithm may solve nonlinear system, nonlinear differential equations; the initial solution is transformed to scattering data (direct scattering transform), the scattering data evolves forward in time (time evolution), and the scattering data reconstructs the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integrable System
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, that its motion is confined to a submanifold of much smaller dimensionality than that of its phase space. Three features are often referred to as characterizing integrable systems: * the existence of a ''maximal'' set of conserved quantities (the usual defining property of complete integrability) * the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic integrability) * the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability) Integrable systems may be seen as very different in qualitative character from more ''generic'' dynamical systems, which are more typically chaotic syste ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Schrödinger Equation
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, planar waveguides and hot rubidium vapors 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 me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave
In physics, mathematics, engineering, and related fields, a wave is a propagating dynamic disturbance (change from List of types of equilibrium, equilibrium) of one or more quantities. ''Periodic waves'' oscillate repeatedly about an equilibrium (resting) value at some frequency. When the entire waveform moves in one direction, it is said to be a travelling wave; by contrast, a pair of superposition principle, superimposed periodic waves traveling in opposite directions makes a ''standing wave''. In a standing wave, the amplitude of vibration has nulls at some positions where the wave amplitude appears smaller or even zero. There are two types of waves that are most commonly studied in classical physics: mechanical waves and electromagnetic waves. In a mechanical wave, Stress (mechanics), stress and Strain (mechanics), strain fields oscillate about a mechanical equilibrium. A mechanical wave is a local deformation (physics), deformation (strain) in some physical medium that propa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martin Kruskal
Martin David Kruskal (; September 28, 1925 – December 26, 2006) 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. He was a student at the University of Chicago and at New York University, where he completed his Ph.D. under Richard Courant in 1952. He spent much of his career at Princeton University, as a research scientist at the Plasma Physics Laboratory starting in 1951, and then as a professor of astronomy (1961), founder and chair of the Program in Applied and Computational Mathematics (1968), and professor of mathematics (1979). He retired from Princeton University in 1989 and joined the mathematics department of Rutgers University, holding the David Hilbert Chair of Mathematics. Apart from serious mathematical work, Kruskal was know ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Tank
A wave tank is a laboratory setup for observing the behavior of surface waves. The typical wave tank is a 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 has a wave-absorbing surface. A similar device is the ripple tank, which is flat and shallow and used for observing patterns of surface waves from above. Wave basin A wave basin is a wave tank which has a width and length of comparable magnitude, often used for testing ships, offshore structures and three-dimensional models of harbors (and their breakwaters). Wave flume A wave flume (or wave channel) is a special sort of wave tank: the width of the flume is much less than its length. The generated waves are therefore – more or less – two-dimensional in a vertical plane (2DV), meaning that the orbital flow velocity component in the direction perpendicular to the flume side wall is much smaller than the other two comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Optics
Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in Nonlinearity, nonlinear media, that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typically observed only at very high light intensities (when the electric field of the light is >108 V/m and thus comparable to the atomic electric field of ~1011 V/m) such as those provided by lasers. Above the Schwinger limit, the vacuum itself is expected to become nonlinear. In nonlinear optics, the superposition principle no longer holds. History The first nonlinear optical effect to be predicted was two-photon absorption, by Maria Goeppert Mayer for her PhD in 1931, but it remained an unexplored theoretical curiosity until 1961 and the almost simultaneous observation of two-photon absorption at Bell Labs and the discovery of second-harmonic generation by Peter Franken ''et al.'' at University of Michigan, both shortly after th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Union Canal (Scotland)
The Union Canal, full name the Edinburgh and Glasgow Union Canal, is a canal in Scotland, running from Falkirk to Edinburgh, constructed to bring minerals, especially coal, to the capital. It was opened in 1822 and was initially successful, but the construction of railways, particularly the Edinburgh and Glasgow Railway, which opened in 1842, diminished its value as a transport medium. It fell into slow commercial decline and was closed to commercial traffic in 1933. It was officially closed in 1965. The canal is listed as three individual scheduled monuments by Historic Scotland according to the three former counties, Midlothian, West Lothian and Stirlingshire, through which it flows. It has benefited from a general revival of interest in canals and, as a result of the Millennium Link, was reopened in 2001 and reconnected to the Forth and Clyde Canal in 2002 by the Falkirk Wheel. It is now in popular use for leisure purposes. History Proposal The canal was conceived ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
