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Inverse scattering transform

Index Inverse scattering transform

In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. [1]

33 relations: Bogomol'nyi–Prasad–Sommerfield bound, Boris Levitan, Camassa–Holm equation, Derivative, Dym equation, Eigenvalues and eigenvectors, Fourier transform, Function (mathematics), If and only if, Integrable system, Integral equation, Ishimori equation, Israel Gelfand, Kadomtsev–Petviashvili equation, Kibibyte, Korteweg–de Vries equation, Lax pair, Magnetic monopole, Marchenko equation, Mathematics, Nonlinear Schrödinger equation, Operator (mathematics), Ordinary differential equation, Partial derivative, Partial differential equation, Physics, Real number, Riemann–Hilbert problem, Schrödinger equation, Sine-Gordon equation, Soliton, Toda lattice, Vladimir Marchenko.

Bogomol'nyi–Prasad–Sommerfield bound

The Bogomol'nyi–Prasad–Sommerfield bound (named after Evgeny Bogomolny, Manoj Prasad, and Charles Sommerfield) is a series of inequalities for solutions of partial differential equations depending on the homotopy class of the solution at infinity.

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Boris Levitan

Boris Levitan (7 June 1914 – 4 April 2004) was a mathematician known in particular for his work on almost periodic functions, and Sturm–Liouville operators, especially, on inverse scattering.

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Camassa–Holm equation

In fluid dynamics, the Camassa–Holm equation is the integrable, dimensionless and non-linear partial differential equation u_t + 2\kappa u_x - u_ + 3 u u_x.

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Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).

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Dym equation

In mathematics, and in particular in the theory of solitons, the Dym equation (HD) is the third-order partial differential equation It is often written in the equivalent form The Dym equation first appeared in Kruskal and is attributed to an unpublished paper by Harry Dym.

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Eigenvalues and eigenvectors

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.

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Fourier transform

The Fourier transform (FT) decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes.

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Function (mathematics)

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.

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If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

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Integrable system

In the context of differential equations to integrate an equation means to solve it from initial conditions.

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Integral equation

In mathematics, an integral equation is an equation in which an unknown function appears under an integral sign.

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Ishimori equation

The Ishimori equation (IE) is a partial differential equation proposed by the Japanese mathematician.

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Israel Gelfand

Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand (ישראל געלפֿאַנד, Изра́иль Моисе́евич Гельфа́нд; – 5 October 2009) was a prominent Soviet mathematician.

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Kadomtsev–Petviashvili equation

In mathematics and physics, the Kadomtsev–Petviashvili equation – or KP equation, named after Boris Borisovich Kadomtsev and Vladimir Iosifovich Petviashvili – is a partial differential equation to describe nonlinear wave motion.

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Kibibyte

The kibibyte is a multiple of the unit byte for quantities of digital information.

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Korteweg–de Vries equation

In mathematics, the Korteweg–de Vries equation (KdV equation for short) is a mathematical model of waves on shallow water surfaces.

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Lax pair

In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the Lax equation.

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Magnetic monopole

A magnetic monopole is a hypothetical elementary particle in particle physics that is an isolated magnet with only one magnetic pole (a north pole without a south pole or vice versa).

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Marchenko equation

In mathematical physics, more specific in the one-dimensional inverse scattering problem, the Marchenko equation (or Gelfand-Levitan-Marchenko equation or GLM equation), named after Israel Gelfand, Boris Levitan and Vladimir Marchenko, is derived by computing the Fourier transform of the scattering relation: K(r,r^\prime) + g(r,r^\prime) + \int_r^ K(r,r^) g(r^,r^\prime) \mathrmr^.

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Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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Nonlinear Schrödinger equation

In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation.

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Operator (mathematics)

In mathematics, an operator is generally a mapping that acts on the elements of a space to produce other elements of the same space.

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Ordinary differential equation

In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and its derivatives.

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Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).

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Partial differential equation

In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.

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Physics

Physics (from knowledge of nature, from φύσις phýsis "nature") is the natural science that studies matterAt the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept: "If, in some cataclysm, all scientific knowledge were to be destroyed one sentence what statement would contain the most information in the fewest words? I believe it is that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another..." and its motion and behavior through space and time and that studies the related entities of energy and force."Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves."Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena.""Physics is the study of your world and the world and universe around you." Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics often enable advances in new technologies. For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.

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Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

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Riemann–Hilbert problem

In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane.

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Schrödinger equation

In quantum mechanics, the Schrödinger equation is a mathematical equation that describes the changes over time of a physical system in which quantum effects, such as wave–particle duality, are significant.

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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.

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Soliton

In mathematics and physics, a soliton is a self-reinforcing solitary wave packet that maintains its shape while it propagates at a constant velocity.

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Toda lattice

The Toda lattice, introduced by, is a simple model for a one-dimensional crystal in solid state physics.

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Vladimir Marchenko

Vladimir Alexandrovich Marchenko (Влади́мир Алекса́ндрович Ма́рченко, Володимир Олександрович Марченко; born July 7, 1922) is a Soviet and Ukrainian mathematician who specializes in mathematical physics.

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Redirects here:

Inverse scattering method, Inverse scattering theory.

References

[1] https://en.wikipedia.org/wiki/Inverse_scattering_transform

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