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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] |
Joseph Valentin Boussinesq
Joseph Valentin Boussinesq (; 13 March 1842 – 19 February 1929) was a French mathematician and physicist who made significant contributions to the theory of hydrodynamics, vibration, light, and heat. Biography From 1872 to 1886, he was appointed professor at Faculty of Sciences of Lille, lecturing differential and integral calculus at Institut industriel du Nord ( École centrale de Lille). From 1896 to his retirement in 1918, he was professor of mechanics at Faculty of Sciences of Paris. John Scott Russell experimentally observed solitary waves in 1834 and reported it during the 1844 Meeting of the British Association for the advancement of science. Subsequently, this was developed into the modern physics of solitons. In 1871, Boussinesq published the first mathematical theory to support Russell's experimental observation, and in 1877 introduced the Korteweg–De Vries equation. In 1876, Lord Rayleigh published his mathematical theory to support Russell's experimental o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Initial Condition
In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). For a system of order ''k'' (the number of time lags in discrete time, or the order of the largest derivative in continuous time) and dimension ''n'' (that is, with ''n'' different evolving variables, which together can be denoted by an ''n''-dimensional coordinate vector), generally ''nk'' initial conditions are needed in order to trace the system's variables forward through time. In both differential equations in continuous time and difference equations in discrete time, initial conditions affect the value of the dynamic variables (state variables) at any future time. In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial valu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dym Equation
In mathematics, and in particular in the theory of solitons, the Dym equation (HD) is the third-order partial differential equation :u_t = u^3u_.\, It is often written in the equivalent form for some function v of one space variable and time : v_t=(v^)_.\, The Dym equation first appeared in Kruskal Martin Kruskal ''Nonlinear Wave Equations''. In Jürgen Moser, editor, Dynamical Systems, Theory and Applications, volume 38 of Lecture Notes in Physics, pages 310–354. Heidelberg. Springer. 1975. and is attributed to an unpublished paper by Harry Dym. The Dym equation represents a system in which dispersion and nonlinearity are coupled together. HD is a completely integrable nonlinear evolution equation that may be solved by means of the inverse scattering transform. It obeys an infinite number of conservation laws; it does not possess the Painlevé property. The Dym equation has strong links to the Korteweg–de Vries equation. C.S. Gardner, J.M. Greene, Kruskal and R.M. Mi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Toda Lattice
The Toda lattice, introduced by , is a simple model for a one-dimensional crystal in solid state physics. It is famous because it is one of the earliest examples of a non-linear completely integrable system. It is given by a chain of particles with nearest neighbor interaction, described by the Hamiltonian :\begin H(p,q) &= \sum_ \left(\frac +V(q(n+1,t)-q(n,t))\right) \end and the equations of motion :\begin \frac p(n,t) &= -\frac = e^ - e^, \\ \frac q(n,t) &= \frac = p(n,t), \end where q(n,t) is the displacement of the n-th particle from its equilibrium position, and p(n,t) is its momentum (mass m=1), and the Toda potential V(r)=e^+r-1. Soliton solutions Soliton solutions are solitary waves spreading in time with no change to their shape and size and interacting with each other in a particle-like way. The general N-soliton solution of the equation is : \begin q_N(n,t)=q_+ + \log \frac , \end where :C_N(n,t)=\Bigg(\frac\Bigg)_, with :\gamma_j(n,t)=\gamma_j\,e^ where \kap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ishimori Equation
The Ishimori equation is a partial differential equation proposed by the Japanese mathematician . Its interest is as the first example of a nonlinear spin-one field model in the plane that is integrable . Equation The Ishimori equation has the form Lax representation The Lax representation of the equation is given by Here the \sigma_i are the Pauli matrices and I is the identity matrix. Reductions The Ishimori equation admits an important reduction: in 1+1 dimensions it reduces to the continuous classical Heisenberg ferromagnet equation (CCHFE). The CCHFE is integrable. Equivalent counterpart The equivalent counterpart of the Ishimori equation is the Davey-Stewartson equation. See also * Nonlinear Schrödinger equation * Heisenberg model (classical) * Spin wave * Landau–Lifshitz model * Soliton * Vortex * Nonlinear systems In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kadomtsev–Petviashvili Equation
In mathematics and physics, the Kadomtsev–Petviashvili equation (often abbreviated as KP equation) is a partial differential equation to describe nonlinear wave motion. Named after Boris Kadomtsev, Boris Borisovich Kadomtsev and Vladimir Iosifovich Petviashvili, the KP equation is usually written as \displaystyle \partial_x(\partial_t u+u \partial_x u+\epsilon^2\partial_u)+\lambda\partial_u=0 where \lambda=\pm 1. The above form shows that the KP equation is a generalization to two spatial dimensions, ''x'' and ''y'', of the one-dimensional Korteweg–de Vries equation, Korteweg–de Vries (KdV) equation. To be physically meaningful, the wave propagation direction has to be not-too-far from the ''x'' direction, i.e. with only slow variations of solutions in the ''y'' direction. Like the KdV equation, the KP equation is completely integrable. It can also be solved using the inverse scattering transform much like the nonlinear Schrödinger equation. In 2002, the regularized version ... [...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] |
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] |
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] |
Fermi–Pasta–Ulam–Tsingou Problem
In physics, the Fermi–Pasta–Ulam–Tsingou (FPUT) problem or formerly the Fermi–Pasta–Ulam problem was the apparent paradox in chaos theory that many complicated enough physical systems exhibited almost exactly periodic behavior – called Fermi–Pasta–Ulam–Tsingou recurrence (or Fermi–Pasta–Ulam recurrence) – instead of the expected ergodic behavior. This came as a surprise, as Enrico Fermi, certainly, expected the system to thermalize in a fairly short time. That is, it was expected for all vibrational modes to eventually appear with equal strength, as per the equipartition theorem, or, more generally, the ergodic hypothesis. Yet here was a system that appeared to evade the ergodic hypothesis. Although the recurrence is easily observed, it eventually became apparent that over much, much longer time periods, the system does eventually thermalize. Multiple competing theories have been proposed to explain the behavior of the system, and it remains a topic of act ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
