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Korteweg–de Vries equation

Index 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. [1]

52 relations: Acoustics, Benjamin–Bona–Mahony equation, Boussinesq approximation (water waves), Bravais lattice, Cnoidal wave, Constant of integration, Constant of motion, Dispersion (water waves), Dispersionless equation, Equations of motion, Euler–Lagrange equation, Fermi–Pasta–Ulam–Tsingou problem, Fifth-order Korteweg–de Vries equation, Function (mathematics), Generalized Korteweg–de Vries equation, Gross–Pitaevskii equation, Huygens–Fresnel principle, Hyperbolic function, Integrable system, Internal wave, Inverse scattering transform, Ion acoustic wave, John Scott Russell, John William Strutt, 3rd Baron Rayleigh, Joseph Valentin Boussinesq, Kadomtsev–Petviashvili equation, Lagrangian (field theory), Lax pair, Mathematical model, Mathematics, Maxima and minima, Method of steepest descent, Modified KdV–Burgers equation, Nonlinear Schrödinger equation, Nonlinear system, Novikov–Veselov equation, Ocean, Ordinary differential equation, Partial derivative, Partial differential equation, Phase velocity, Plasma (physics), Q-analog, Real number, Riemann–Hilbert problem, Seventh-order Korteweg–de Vries equation, Soliton, Springer Science+Business Media, Sturm–Liouville theory, Ursell number, ..., Vector soliton, Wolfram Demonstrations Project. Expand index (2 more) »

Acoustics

Acoustics is the branch of physics that deals with the study of all mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound.

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Benjamin–Bona–Mahony equation

The Benjamin–Bona–Mahony equation (or BBM equation) – also known as the regularized long-wave equation (RLWE) – is the partial differential equation This equation was studied in as an improvement of the Korteweg–de Vries equation (KdV equation) for modeling long surface gravity waves of small amplitude – propagating uni-directionally in 1+1 dimensions.

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Boussinesq approximation (water waves)

In fluid dynamics, the Boussinesq approximation for water waves is an approximation valid for weakly non-linear and fairly long waves.

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Bravais lattice

In geometry and crystallography, a Bravais lattice, named after, is an infinite array of discrete points in three dimensional space generated by a set of discrete translation operations described by: where ni are any integers and ai are known as the primitive vectors which lie in different directions and span the lattice.

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Cnoidal wave

In fluid dynamics, a cnoidal wave is a nonlinear and exact periodic wave solution of the Korteweg–de Vries equation.

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Constant of integration

In calculus, the indefinite integral of a given function (i.e., the set of all antiderivatives of the function) on a connected domain is only defined up to an additive constant, the constant of integration.

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Constant of motion

In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion.

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Dispersion (water waves)

In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds.

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Dispersionless equation

Dispersionless (or quasi-classical) limits of integrable partial differential equations (PDE) arise in various problems of mathematics and physics and have been intensively studied in recent literature (see, f.i., -). They typically arise when considering slowly modulated long waves of an integrable dispersive PDE system.

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Equations of motion

In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time.

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Euler–Lagrange equation

In the calculus of variations, the Euler–Lagrange equation, Euler's equation, or Lagrange's equation (although the latter name is ambiguous—see disambiguation page), is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary.

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Fermi–Pasta–Ulam–Tsingou problem

In physics, the Fermi–Pasta–Ulam–Tsingou 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 ergodic behavior.

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Fifth-order Korteweg–de Vries equation

A fifth-order Korteweg–de Vries (KdV) equation is a nonlinear partial differential equation in 1+1 dimensions related to the Korteweg–de Vries equation.

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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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Generalized Korteweg–de Vries equation

In mathematics a generalized Korteweg–de Vries equation is the nonlinear partial differential equation The function f is sometimes taken to be f(u).

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Gross–Pitaevskii equation

The Gross–Pitaevskii equation (GPE, named after Eugene P. Gross and Lev Petrovich Pitaevskii) describes the ground state of a quantum system of identical bosons using the Hartree–Fock approximation and the pseudopotential interaction model.

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Huygens–Fresnel principle

The Huygens–Fresnel principle (named after Dutch physicist Christiaan Huygens and French physicist Augustin-Jean Fresnel) is a method of analysis applied to problems of wave propagation both in the far-field limit and in near-field diffraction.

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Hyperbolic function

In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions.

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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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Internal wave

Internal waves are gravity waves that oscillate within a fluid medium, rather than on its surface.

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Inverse scattering transform

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

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Ion acoustic wave

In plasma physics, an ion acoustic wave is one type of longitudinal oscillation of the ions and electrons in a plasma, much like acoustic waves traveling in neutral gas.

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John Scott Russell

John Scott Russell FRSE FRS (9 May 1808, Parkhead, Glasgow – 8 June 1882, Ventnor, Isle of Wight) was a Scottish civil engineer, naval architect and shipbuilder who built the Great Eastern in collaboration with Isambard Kingdom Brunel.

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John William Strutt, 3rd Baron Rayleigh

John William Strutt, 3rd Baron Rayleigh, (12 November 1842 – 30 June 1919) was a physicist who, with William Ramsay, discovered argon, an achievement for which he earned the Nobel Prize for Physics in 1904.

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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.

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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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Lagrangian (field theory)

Lagrangian field theory is a formalism in classical field theory.

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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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Mathematical model

A mathematical model is a description of a system using mathematical concepts and language.

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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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Maxima and minima

In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire domain of a function (the global or absolute extrema).

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Method of steepest descent

In mathematics, the method of steepest descent or stationary-phase method or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase.

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Modified KdV–Burgers equation

The modified KdV–Burgers equation is a nonlinear partial differential equation.

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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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Nonlinear system

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.

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Novikov–Veselov equation

In mathematics, the Novikov–Veselov equation (or Veselov–Novikov equation) is a natural (2+1)-dimensional analogue of the Korteweg–de Vries (KdV) equation.

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Ocean

An ocean (the sea of classical antiquity) is a body of saline water that composes much of a planet's hydrosphere.

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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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Phase velocity

The phase velocity of a wave is the rate at which the phase of the wave propagates in space.

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Plasma (physics)

Plasma (Henry George Liddell, Robert Scott, A Greek English Lexicon, on Perseus) is one of the four fundamental states of matter, and was first described by chemist Irving Langmuir in the 1920s.

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Q-analog

In mathematics, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as.

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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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Seventh-order Korteweg–de Vries equation

The seventh-order KdV equation is a nonlinear partial differential equation in 1+1 dimensions related to the KdV equation.

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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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Springer Science+Business Media

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

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Sturm–Liouville theory

In mathematics and its applications, a classical Sturm–Liouville theory, named after Jacques Charles François Sturm (1803–1855) and Joseph Liouville (1809–1882), is the theory of a real second-order linear differential equation of the form where y is a function of the free variable x. Here the functions p(x), q(x), and w(x) > 0 are specified at the outset.

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Ursell number

In fluid dynamics, the Ursell number indicates the nonlinearity of long surface gravity waves on a fluid layer.

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Vector soliton

In physical optics or wave optics, a vector soliton is a solitary wave with multiple components coupled together that maintains its shape during propagation.

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Wolfram Demonstrations Project

The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields.

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References

[1] https://en.wikipedia.org/wiki/Korteweg–de_Vries_equation

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