Category: Exactly solvable models

Two-dimensional critical Ising model
The two-dimensional critical Ising model is the critical limit of the Ising model in two dimensions. It is a two-dimensional conformal field theory whose symmetry algebra is the Virasoro algebra with
Boolean network
A Boolean network consists of a discrete set of boolean variables each of which has a Boolean function (possibly different for each variable) assigned to it which takes inputs from a subset of those v
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. It was originally introduce
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
Bretherton equation
In mathematics, the Bretherton equation is a nonlinear partial differential equation introduced by Francis Bretherton in 1964: with integer and While and denote partial derivatives of the scalar field
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
Kundu equation
The Kundu equation is a general form of integrable system that is gauge-equivalent to the mixed nonlinear Schrödinger equation. It was proposed by Anjan Kundu as with arbitrary function and the subscr
Hard hexagon model
In statistical mechanics, the hard hexagon model is a 2-dimensional lattice model of a gas, where particles are allowed to be on the vertices of a triangular lattice but no two particles may be adjace
Corner transfer matrix
In statistical mechanics, the corner transfer matrix describes the effect of adding a quadrant to a lattice. Introduced by Rodney Baxter in 1968 as an extension of the Kramers-Wannier row-to-row trans
Wess–Zumino–Witten model
In theoretical physics and mathematics, a Wess–Zumino–Witten (WZW) model, also called a Wess–Zumino–Novikov–Witten model, is a type of two-dimensional conformal field theory named after Julius Wess, B
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
Nonlinear partial differential equation
In mathematics and physics, a nonlinear partial differential equation is a partial differential equation with nonlinear terms. They describe many different physical systems, ranging from gravitation t
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
Topological quantum number
In physics, a topological quantum number (also called topological charge) is any quantity, in a physical theory, that takes on only one of a discrete set of values, due to topological considerations.
Bogdanov map
In dynamical systems theory, the Bogdanov map is a chaotic 2D map related to the Bogdanov–Takens bifurcation. It is given by the transformation: The Bogdanov map is named after Rifkat Bogdanov.
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
Rosenau–Hyman equation
The Rosenau–Hyman equation or K(n,n) equation is a KdV-like equation having compacton solutions. This nonlinear partial differential equation is of the form The equation is named after Philip Rosenau
Ice-type model
In statistical mechanics, the ice-type models or six-vertex models are a family of vertex models for crystal lattices with hydrogen bonds. The first such model was introduced by Linus Pauling in 1935
Schwinger model
In physics, the Schwinger model, named after Julian Schwinger, is the model describing 1+1D (1 spatial dimension + time) Lorentzian quantum electrodynamics which includes electrons, coupled to photons
Thirring–Wess model
The Thirring–Wess model or Vector Meson model is an exactly solvablequantum field theory describing the interaction of a Dirac field with a vector field in dimension two.
Chiral Potts curve
The chiral Potts curve is an algebraic curve defined over the complex numbers that occurs in the study of the chiral Potts model of statistical mechanics. For an integer N, the parameters in the Boltz
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
KdV hierarchy
In mathematics, the KdV hierarchy is an infinite sequence of partial differential equations which starts with the Korteweg–de Vries equation.
Thirring model
The Thirring model is an exactly solvable quantum field theory which describes the self-interactions of a Dirac field in (1+1) dimensions.
Kramers–Wannier duality
The Kramers–Wannier duality is a symmetry in statistical physics. It relates the free energy of a two-dimensional square-lattice Ising model at a low temperature to that of another Ising model at a hi
In representation theory, a Yangian is an infinite-dimensional Hopf algebra, a type of a quantum group. Yangians first appeared in physics in the work of Ludvig Faddeev and his school in the late 1970
Bethe ansatz
In physics, the Bethe ansatz is an ansatz method for finding the exact wavefunctions of certain one-dimensional quantum many-body models. It was invented by Hans Bethe in 1931 to find the exact eigenv
Quantum inverse scattering method
In quantum physics, the quantum inverse scattering method is a method for solving integrable models in 1+1 dimensions, introduced by L. D. Faddeev in 1979. Sklyanin, E. K. (1992). "Quantum Inverse Sca
Baker's map
In dynamical systems theory, the baker's map is a chaotic map from the unit square into itself. It is named after a kneading operation that bakers apply to dough: the dough is cut in half, and the two
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. Lax pairs
Z N model
The model (also known as the clock model) is a simplified statistical mechanical spin model. It is a generalization of the Ising model. Although it can be defined on an arbitrary graph, it is integrab
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
Spherical model
The spherical model is a model of ferromagnetism similar to the Ising model, which was solved in 1952 by T. H. Berlin and M. Kac. It has the remarkable property that for linear dimension d greater tha
Woods–Saxon potential
The Woods–Saxon potential is a mean field potential for the nucleons (protons and neutrons) inside the atomic nucleus, which is used to describe approximately the forces applied on each nucleon, in th
List of integrable models
This is a list of integrable models as well as classes of integrable models in physics.
Square lattice Ising model
In statistical mechanics, the two-dimensional square lattice Ising model is a simple lattice model of interacting magnetic spins. The model is notable for having nontrivial interactions, yet having an
Eight-vertex model
In statistical mechanics, the eight-vertex model is a generalisation of the ice-type (six-vertex) models; it was discussed by Sutherland, and Fan & Wu, and solved by Baxter in the zero-field case.
Yang–Baxter equation
In physics, the Yang–Baxter equation (or star–triangle relation) is a which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, parti
Arnold's cat map
In mathematics, Arnold's cat map is a chaotic map from the torus into itself, named after Vladimir Arnold, who demonstrated its effects in the 1960s using an image of a cat, hence the name. Thinking o
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 Borisovich Ka
Quantum affine algebra
In mathematics, a quantum affine algebra (or affine quantum group) is a Hopf algebra that is a q-deformation of the universal enveloping algebra of an affine Lie algebra. They were introduced independ
Ising model
The Ising model (German pronunciation: [ˈiːzɪŋ]) (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statis
Potts model
In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. By studying the Potts model, one may gain insight into the beh
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
Gingerbreadman map
In dynamical systems theory, the Gingerbreadman map is a chaotic two-dimensional map. It is given by the piecewise linear transformation:
Minimal model (physics)
In theoretical physics, a minimal model or Virasoro minimal model is a two-dimensional conformal field theory whose spectrum is built from finitely many irreducible representations of the Virasoro alg
Domino tiling
In geometry, a domino tiling of a region in the Euclidean plane is a tessellation of the region by dominoes, shapes formed by the union of two unit squares meeting edge-to-edge. Equivalently, it is a
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
Kuramoto model
The Kuramoto model (or Kuramoto–Daido model), first proposed by Yoshiki Kuramoto (蔵本 由紀, Kuramoto Yoshiki), is a mathematical model used to describing synchronization. More specifically, it is a model