This is a list of integrable models as well as classes of

integrable model 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 sufficiently many conserved quantities, or first in ...

s in physics.

Integrable models in 1+1 dimensions

In classical and quantum field theory: *free boson *free fermion *

sine-Gordon model 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 introduced by in the course of study of surfa ...

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. Definition The Thirring model is given by the Lagrangian density : \mathcal= \overline(i\partial\!\!\!/- ...

* sinh-Gordon model *

Liouville field theory In physics, Liouville field theory (or simply Liouville theory) is a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation. Liouville theory is defined for all complex values of the ...

* Bullough–Dodd model *

Dym equation In mathematics, and in particular in the theory of solitons, the Dym equation (HD) is the third-order partial differential equation :u_t = u^3u_.\, It is often written in the equivalent form for some function v of one space variable and time ...

Calogero–Degasperis–Fokas equation In mathematics, the Calogero–Degasperis–Fokas equation is the nonlinear partial differential equation :\displaystyle u_t=u_-\fracu_x^3 + u_x\left(Ae^u+Be^\right). This equation was named after F. Calogero, A. Degasperis, and A. Fokas Athan ...

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 = 2 u_x u_ + u u_. \, The equation was introduced by Roberto Camassa and Darryl Ho ...

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 Vladimir Gershonovich Drinfeld ( uk, Володи́мир Ге́ршонович Дрін ...

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 equation is :u_t+uu_x+Hu_=0 where ''H'' i ...

*SS model *sausage model * Toda field theories *O(''N'')-symmetric non-linear sigma models *

Ernst equation In mathematics, the Ernst equation is an integrable non-linear partial differential equation, named after the American physicist . The Ernst equation The equation reads: \Re(u)(u_+u_r/r+u_) = (u_r)^2+(u_z)^2. For its Lax pair and other feat ...

* massless Schwinger model *supersymmetric sine-Gordon model *supersymmetric sinh-Gordon model * conformal minimal models *critical Ising model *tricritical Ising model *3-state

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 behaviour of ferromagnets and certain other phen ...

*various perturbations of conformal minimal models *superconformal minimal models *

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, Bruno Zumino, Sergei Novikov and E ...

Nonlinear Schroedinger equation 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. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...

* Korteweg–de Vries equation *

modified Korteweg–de Vries equation Modified may refer to: * ''Modified'' (album), the second full-length album by Save Ferris *Modified racing, or "Modifieds", an American automobile racing genre See also * Modification (disambiguation) Modification may refer to: * Modification ...

* Gardner equation * Gibbons–Tsarev equation * Hunter–Saxton equation *

Kaup–Kupershmidt equation The Kaup–Kupershmidt equation (named after David J. Kaup and Boris Abram Kupershmidt) is the nonlinear fifth-order partial differential equation :u_t = u_+10u_u+25u_u_x+20u^2u_x = \frac16 (6u_+60uu_+45u_x^2+40u^3)_x. It is the first equation in ...

* XXX spin chain * XXZ spin chain * XYZ spin chain * 6-vertex model *

8-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. Description As with the ice-type models, the ei ...

*Kondo Model *Anderson impurity model *Chiral Gross–Neveu model

Integrable models in 2+1 dimensions

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 integrable . Equation The Ishimori equation has the for ...

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 Kadomtsev and Vladimir Iosifovich Petviashvi ...

Landau–Lifshitz–Gilbert equation In physics, the Landau–Lifshitz–Gilbert equation, named for Lev Landau, Evgeny Lifshitz, and T. L. Gilbert, is a name used for a differential equation describing the precessional motion of magnetization in a solid. It is a modification by Gi ...

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 equation, Korteweg–de Vries (KdV) equation. Unlike another (2+1)-dimensional analogue of KdV, the Kad ...

Integrable models in 3+1 dimensions

Self-dual Yang–Mills equations In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the d ...

* Systems with contact Lax pairs

In quantum mechanics

harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \v ...

hydrogen atom A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen cons ...

Hooke's atom Hooke's atom, also known as harmonium or hookium, refers to an artificial helium-like atom where the Coulombic electron-nucleus interaction potential is replaced by a harmonic potential. This system is of significance as it is, for certain value ...

(Hookium) * Ruijsenaars–Schneider models * Calogero–Moser models *

Inverse square root potential Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when a ...

* Lambert-W step-potential * Multistate Landau–Zener Models

References

{{Reflist Integrable systems Exactly solvable models