# Category: Integrable systems

Toda field theory
In mathematics and physics, specifically the study of field theory and partial differential equations, a Toda field theory, named after Morikazu Toda, is specified by a choice of Kac–Moody algebra and
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. Several existe
Frobenius manifold
In the mathematical field of differential geometry, a Frobenius manifold, introduced by Dubrovin, is a flat Riemannian manifold with a certain compatible multiplicative structure on the tangent space.
Nahm equations
In differential geometry and gauge theory, the Nahm equations are a system of ordinary differential equations introduced by Werner Nahm in the context of the Nahm transform – an alternative to Ward's
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 literatur
Kaup–Kupershmidt equation
The Kaup–Kupershmidt equation (named after David J. Kaup and Boris Abram Kupershmidt) is the nonlinear fifth-order partial differential equation It is the first equation in a hierarchy of integrable e
Ward's conjecture
In mathematics, Ward's conjecture is the conjecture made by Ward that "many (and perhaps all?) of the ordinary and partial differential equations that are regarded as being integrable or solvable may
Toda lattice
The Toda lattice, introduced by Morikazu Toda, 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 completel
Tau function (integrable systems)
Tau functions are an important ingredient in the modern theory of integrable systems, and have numerous applications in a variety of other domains. They were originally introduced by Ryogo Hirota in h
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
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.It was introduced by and . The Benjamin–Ono e
Integrable algorithm
Integrable algorithms are numerical algorithms that rely on basic ideas from the mathematical theory of integrable systems.
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 ind
Bäcklund transform
In mathematics, Bäcklund transforms or Bäcklund transformations (named after the Swedish mathematician Albert Victor Bäcklund) relate partial differential equations and their solutions. They are an im
Bullough–Dodd model
The Bullough–Dodd model is an integrable model in 1+1-dimensional quantum field theory introduced by Robin Bullough and Roger Dodd. ItsLagrangian density is where is a mass parameter, is the coupling
Calogero–Degasperis–Fokas equation
In mathematics, the Calogero–Degasperis–Fokas equation is the nonlinear partial differential equation This equation was named after F. Calogero, , and A. Fokas.
W-algebra
In conformal field theory and representation theory, a W-algebra is an associative algebra that generalizes the Virasoro algebra. W-algebras were introduced by Alexander Zamolodchikov, and the name "W
Gelfand–Zeitlin integrable system
In mathematics, the Gelfand–Zeitlin system (also written Gelfand–Zetlin system, Gelfand–Cetlin system, Gelfand–Tsetlin system) is an integrable system on conjugacy classes of Hermitian matrices. It wa
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 n
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 suff
Ernst equation
In mathematics, the Ernst equation is an integrable non-linear partial differential equation, named after the American physicist .
Liouville–Arnold theorem
In dynamical systems theory, the Liouville–Arnold theorem states that if, in a Hamiltonian dynamical system with n degrees of freedom, there are also n independent,Poisson commuting first integrals of
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 integrab
Gardner equation
The Gardner equation is an integrable nonlinear partial differential equation introduced by the mathematician Clifford Gardner in 1968 to generalize KdV equation and modified KdV equation. The Gardner
Camassa–Holm equation
In fluid dynamics, the Camassa–Holm equation is the integrable, dimensionless and non-linear partial differential equation The equation was introduced by and Darryl Holm as a bi-Hamiltonian model for
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 for some function v of on
Superintegrable Hamiltonian system
In mathematics, a superintegrable Hamiltonian system is a Hamiltonian system on a -dimensional symplectic manifold for which the following conditions hold: (i) There exist independent integrals of mot
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. It is a system of partial dif
AKNS system
In mathematics, the AKNS system is an integrable system of partial differential equations, introduced by and named after Mark J. Ablowitz, David J. Kaup, Alan C. Newell, and Harvey Segur from their pu
Drinfeld–Sokolov–Wilson equation
The Drinfeld–Sokolov–Wilson (DSW) equations are an integrable system of two coupled nonlinear partial differential equations proposed by Vladimir Drinfeld and Vladimir Sokolov, and independently by Ge
List of integrable models
This is a list of integrable models as well as classes of integrable models in physics.
DBAR problem
The DBAR problem, or the -problem, is the problem of solving the differential equation for the function , where is assumed to be known and is a complex number in a domain . The operator is called the
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 Borisovich Ka
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 F
Unnormalized modified KdV equation
The unnormalized modified Korteweg–de Vries (KdV) equation is an integrable nonlinear partial differential equation： where is an arbitrary (nonzero) constant. See also Korteweg–de Vries equation. This
Korteweg–De Vries equation
In mathematics, the Korteweg–De Vries (KdV) equation is a mathematical model of waves on shallow water surfaces. It is particularly notable as the prototypical example of an exactly solvable model, th
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. Unlike another (2+1)-dimensional analogue of