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Soliton
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. 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. ascribe three properties to solitons: # They are of permanent form; # They are localized within a region; # They can interact with other solitons, and emerge from the collision unchanged, e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Soliton Hydro
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. 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. ascribe three properties to solitons: # They are of permanent form; # They are localized within a region; # They can interact with other solitons, and emerge from the collision unchanged, e ... [...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 linear effects in the medium. There are two main kinds of solitons: * spatial solitons: the nonlinear effect can balance the diffraction. 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 some consider ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sine-Gordon Equation
The sine-Gordon equation is a nonlinear hyperbolic partial differential equation in 1 + 1 dimensions involving the d'Alembert operator and the sine of the unknown function. It was originally introduced by in the course of study of surfaces of constant negative curvature as the Gauss–Codazzi equation for surfaces of curvature −1 in 3-space, and 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. Origin of the equation and its name There are two equivalent forms of the sine-Gordon equation. In the (real) ''space-time coordinates'', denoted (''x'', ''t''), the equation reads: : \varphi_ - \varphi_ + \sin\varphi = 0, where partial derivatives are denoted by subscripts. Passing to the light-cone coordinates (''u'', ''v''), akin to ''asymptotic coordinates'' where : u = \frac, \quad v = \frac, the equation takes the ... [...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 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. Unl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Russell (engineer)
John Scott Russell FRSE FRS FRSA (9 May 1808, Parkhead, Glasgow – 8 June 1882, Ventnor, Isle of Wight) 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. Early life John Russell was born on 9 May 1808 in Parkhead, Glasgow, 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 the Leith Mechanics' Institute, achieving the hig ... [...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 ''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 the constru ... [...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, 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. 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 mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exactly Solvable Model
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. 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 m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Light Bullet
Light bullets are localized pulses of electromagnetic energy that can travel through a medium and retain their spatiotemporal shape in spite of diffraction and dispersion which tend to spread the pulse. This is made possible by a balance between the non-linear self-focusing and spreading effects brought about by the medium in which the pulse beam propagates. Prediction and Discovery Light bullets were predicted and so termed by Yaron Silberberg in 1990, and demonstrated the following decade. Comparison with solitons Spatial and temporal stability which are the characteristics of a soliton have been achieved in light bullets using alternative refractive index models. An experiment which exploited the discrete spreading and self-focusing effects on 170-femtosecond pulses at 1550-nanometre wavelengths by a two-dimensional hexagonal array of silica waveguides reported a spatial profiles stationary for about twice as far as it would be in linear propagation and temporal profile about ni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinearity
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems. Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Scattering Transform
In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. 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 inverse scattering transform may be applied to many of the so-called exactly solvable models, that is to say completely integrable infinite dimensional systems. Overview The inverse scattering transform was first introduced by for the Korteweg–de Vries equation, and soon extended to the nonlinear Schrödinger equation, the Sine-Gordon equat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Internal Wave
Internal waves are gravity waves that oscillate within a fluid medium, rather than on its surface. To exist, the fluid must be stratified: the density must change (continuously or discontinuously) with depth/height due to changes, for example, in temperature and/or salinity. If the density changes over a small vertical distance (as in the case of the thermocline in lakes and oceans or an atmospheric inversion), the waves propagate horizontally like surface waves, but do so at slower speeds as determined by the density difference of the fluid below and above the interface. If the density changes continuously, the waves can propagate vertically as well as horizontally through the fluid. Internal waves, also called internal gravity waves, go by many other names depending upon the fluid stratification, generation mechanism, amplitude, and influence of external forces. If propagating horizontally along an interface where the density rapidly decreases with height, they are specifically ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
