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Integrable system

Index Integrable system

In the context of differential equations to integrate an equation means to solve it from initial conditions. [1]

79 relations: Action (physics), Action-angle coordinates, AKNS system, Algebraic geometry, Anatoly Fomenko, Angular momentum, Benjamin–Ono equation, Bethe ansatz, Boussinesq approximation (water waves), Canonical coordinates, Canonical transformation, Central force, Chaos theory, Conserved quantity, Constant of motion, Davey–Stewartson equation, Differential equation, Donaldson theory, Dym equation, Dynamical system, Exact solutions of classical central-force problems, Exterior derivative, Ferdinand Georg Frobenius, Finite difference, Flow (mathematics), Foliation, Frobenius theorem (differential topology), Gennadi Sardanashvily, Gert Sabidussi, Hamiltonian mechanics, Harmonic oscillator, Heisenberg model (quantum), Herbert Goldstein, Hilbert space, Hitchin system, Hubbard model, Initial condition, Integrability conditions for differential systems, Integral, Integral equation, Inverse scattering transform, Ishimori equation, Isotropic quadratic form, John Harnad, Joseph Liouville, Kadomtsev–Petviashvili equation, Korteweg–de Vries equation, Lagrange, Euler, and Kovalevskaya tops, Landau–Lifshitz model, Lieb–Liniger model, ..., Louis Armstrong, Ludvig Faddeev, Michèle Audin, Newton's law of universal gravitation, Nigel Hitchin, Non-linear sigma model, Nonlinear Schrödinger equation, Phase space, Physical Review A, Poisson bracket, Quantum inverse scattering method, Riemann–Hilbert problem, Self-adjoint operator, Separation of variables, Sine-Gordon equation, Soliton, Spin wave, Submanifold, Superintegrable Hamiltonian system, Swinging Atwood's machine, Symplectic geometry, Symplectic manifold, Toda lattice, Torus, Valentin Afraimovich, Vladimir Arnold, Vladimir Korepin, Volterra lattice, Yang–Baxter equation. Expand index (29 more) »

Action (physics)

In physics, action is an attribute of the dynamics of a physical system from which the equations of motion of the system can be derived.

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Action-angle coordinates

In classical mechanics, action-angle coordinates are a set of canonical coordinates useful in solving many integrable systems.

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AKNS system

In mathematics, the AKNS system is an integrable system of partial differential equations, introduced by and named after.

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Algebraic geometry

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.

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Anatoly Fomenko

Anatoly Timofeevich Fomenko (Анато́лий Тимофе́евич Фоме́нко) (born 13 March 1945 in Stalino, USSR) is a Soviet and Russian mathematician, professor at Moscow State University, well known as a topologist, and a member of the Russian Academy of Sciences.

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Angular momentum

In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of linear momentum.

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Benjamin–Ono equation

In mathematics, the Benjamin–Ono equation is a nonlinear partial integro-differential equation that describes one-dimensional internal waves in deep water.

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Bethe ansatz

In physics, the Bethe ansatz is an ansatz method for finding the exact solutions of certain one-dimensional quantum many-body models.

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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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Canonical coordinates

In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time.

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Canonical transformation

In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates that preserves the form of Hamilton's equations.

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Central force

In classical mechanics, a central force on an object is a force that is directed along the line joining the object and the origin: where \scriptstyle \vec is the force, F is a vector valued force function, F is a scalar valued force function, r is the position vector, ||r|| is its length, and \scriptstyle \hat.

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Chaos theory

Chaos theory is a branch of mathematics focusing on the behavior of dynamical systems that are highly sensitive to initial conditions.

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Conserved quantity

In mathematics, a conserved quantity of a dynamical system is a function of the dependent variables whose value remains constant along each trajectory of the system.

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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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Davey–Stewartson equation

In fluid dynamics, the Davey–Stewartson equation (DSE) was introduced in a paper by to describe the evolution of a three-dimensional wave-packet on water of finite depth.

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Differential equation

A differential equation is a mathematical equation that relates some function with its derivatives.

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Donaldson theory

Donaldson theory is the study of the topology of smooth 4-manifolds using moduli spaces of anti-self-dual instantons.

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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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Dynamical system

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space.

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Exact solutions of classical central-force problems

In the classical central-force problem of classical mechanics, some potential energy functions V(r) produce motions or orbits that can be expressed in terms of well-known functions, such as the trigonometric functions and elliptic functions.

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Exterior derivative

On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree.

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Ferdinand Georg Frobenius

Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory.

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Finite difference

A finite difference is a mathematical expression of the form.

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Flow (mathematics)

In mathematics, a flow formalizes the idea of the motion of particles in a fluid.

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Foliation

In mathematics, a foliation is a geometric tool for understanding manifolds.

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Frobenius theorem (differential topology)

In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an underdetermined system of first-order homogeneous linear partial differential equations.

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Gennadi Sardanashvily

Gennadi Sardanashvily (Генна́дий Алекса́ндрович Сарданашви́ли; March 13, 1950 - September 1, 2016) was a theoretical physicist, a principal research scientist of Moscow State University.

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Gert Sabidussi

Gert Sabidussi (born 28 October 1929 in Graz) is an Austrian mathematician specializing in combinatorics and graph theory.

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Hamiltonian mechanics

Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics and predicts the same outcomes as non-Hamiltonian classical mechanics.

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Harmonic oscillator

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: where k is a positive constant.

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Heisenberg model (quantum)

The Heisenberg model is a statistical mechanical model used in the study of critical points and phase transitions of magnetic systems, in which the spins of the magnetic systems are treated quantum mechanically.

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Herbert Goldstein

Herbert Goldstein (June 26, 1922 – January 12, 2005) was an American physicist and the author of the standard graduate textbook Classical Mechanics.

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Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

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Hitchin system

In mathematics, the Hitchin integrable system is an integrable system depending on the choice of a complex reductive group and a compact Riemann surface, introduced by Nigel Hitchin in 1987.

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Hubbard model

The Hubbard model is an approximate model used, especially in solid-state physics, to describe the transition between conducting and insulating systems.

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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.

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Integrability conditions for differential systems

In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms.

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Integral

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.

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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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Inverse scattering transform

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

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Ishimori equation

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

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Isotropic quadratic form

In mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which the form evaluates to zero.

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John Harnad

John Harnad (born Hernád János) is a Hungarian-born mathematical physicist.

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Joseph Liouville

Joseph Liouville FRS FRSE FAS (24 March 1809 – 8 September 1882) was a French 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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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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Lagrange, Euler, and Kovalevskaya tops

In classical mechanics, the precession of a rigid body such as a top under the influence of gravity is not, in general, an integrable problem.

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Landau–Lifshitz model

In solid-state physics, the Landau–Lifshitz equation (LLE), named for Lev Landau and Evgeny Lifshitz, is a partial differential equation describing time evolution of magnetism in solids, depending on 1 time variable and 1, 2, or 3 space variables.

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Lieb–Liniger model

The Lieb–Liniger model describes a gas of particles moving in one dimension and satisfying Bose–Einstein statistics.

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Louis Armstrong

Louis Daniel Armstrong (August 4, 1901 – July 6, 1971), nicknamed Satchmo, Satch, and Pops, was an American trumpeter, composer, singer and occasional actor who was one of the most influential figures in jazz.

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Ludvig Faddeev

Ludvig Dmitrievich Faddeev (also Ludwig Dmitriyevich; Лю́двиг Дми́триевич Фадде́ев; 23 March 1934 – 26 February 2017) was a Soviet and Russian theoretical physicist and mathematician.

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Michèle Audin

Michèle Audin is a French mathematician, writer, and a former professor.

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Newton's law of universal gravitation

Newton's law of universal gravitation states that a particle attracts every other particle in the universe with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.

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Nigel Hitchin

Nigel James Hitchin FRS (born 2 August 1946) a British mathematician working in the fields of differential geometry, algebraic geometry, and mathematical physics.

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Non-linear sigma model

In quantum field theory, a nonlinear σ model describes a scalar field which takes on values in a nonlinear manifold called the target manifold T.

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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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Phase space

In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space.

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Physical Review A

Physical Review A (also known as PRA) is a monthly peer-reviewed scientific journal published by the American Physical Society covering atomic, molecular, and optical physics and quantum information.

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Poisson bracket

In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system.

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Quantum inverse scattering method

The quantum inverse scattering method relates two different approaches.

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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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Self-adjoint operator

In mathematics, a self-adjoint operator on a finite-dimensional complex vector space V with inner product \langle\cdot,\cdot\rangle is a linear map A (from V to itself) that is its own adjoint: \langle Av,w\rangle.

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Separation of variables

In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.

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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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Spin wave

Spin waves are propagating disturbances in the ordering of magnetic materials.

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Submanifold

In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties.

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Superintegrable Hamiltonian system

In mathematics, a superintegrable Hamiltonian system is a Hamiltonian system on a 2n-dimensional symplectic manifold for which the following conditions hold: (i) There exist k>n independent integrals F_i of motion.

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Swinging Atwood's machine

The swinging Atwood's machine (SAM) is a mechanism that resembles a simple Atwood's machine except that one of the masses is allowed to swing in a two-dimensional plane, producing a dynamical system that is chaotic for some system parameters and initial conditions.

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Symplectic geometry

Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form.

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Symplectic manifold

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form.

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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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Torus

In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.

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Valentin Afraimovich

Valentin Afraimovich (Валентин Сендерович Афраймович, 2 April 1945, Kirov, Kirov Oblast, USSR – 21 February 2018, Nizhny Novgorod, Russia) was a Soviet, Russian and Mexican mathematician.

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Vladimir Arnold

Vladimir Igorevich Arnold (alternative spelling Arnol'd, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician.

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Vladimir Korepin

Vladimir Korepin (born 1951) is a professor at the C. N. Yang Institute of Theoretical Physics of the Stony Brook University.

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Volterra lattice

In mathematics, the Volterra lattice, also known as the discrete KdV equation, the Kac–van Moerbeke lattice, and the Langmuir lattice, is a system of ordinary differential equations with variables indexed by some of the points of a 1-dimensional lattice.

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Yang–Baxter equation

In physics, the Yang–Baxter equation (or star-triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics.

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References

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

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