In theoretical physics, an exactly solvable model or integrable model refers to a physical model, a physical theory, or set of differential equations whose exact solution may be calculated analytically in terms of elementary or special functions; the adjective integrable therefore implies solvablility. Models in statistical mechanics are considered to be completely solved if one has an exact expression for the partition function as a function of the parameters, or the full set of correlation functions. Theoretical physics employs mathematical models and abstractions of physics, as opposed to experimental processes, in an attempt to understand nature. ... Theoretical physics attempts to understand the world by making a model of reality, used for rationalizing, explaining, predicting physical phenomena through a physical theory. There are three types of theories in physics; mainstream theories, proposed theories and fringe theories. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In mathematics, several functions are important enough to deserve their own name. ... Integrability is a mathematical concept used in different areas. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... In number theory, see Partition function (number theory) In statistical mechanics, see Partition function (statistical mechanics) In quantum field theory, see Partition function (quantum field theory) In game theory, see Partition function (game theory) This is a disambiguation page — a navigational aid which lists other pages that might otherwise... For stochastic processes, including those that arise in statistical mechanics and Euclidean quantum field theory, a correlation function is the correlation between random variables at two different points in space or time. ...

The term exactly solvable model is usually reserved for more complex, and almost always non-linear systems, rather than applying broadly to all possible integrable 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. ...

The study of completely integrable non-linear partial differential equation began with the discovery and study of solitons by Zabusky and Kruskal in the Korteweg-de Vries equation (KdV) in 1965. Arising as an approximate model in many physical systems, the KdV serves as the prototypical example of an exactly solvable model, and continues to be the best known and the most studied partial differential equation that is completely integrable. In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... The Korteweg-de Vries equation (KdV equation for short) is the following partial differential equation for a function φ of two real variables, x and t: Its solutions clump up into solitons. ...

Exactly solvable models show up in a wide range of applications in engineering, numerical analysis, and mathematical physics, as well as economics, and mathematical biology. One practical example is the Manakov model of the propagation of solitons in fiber optics; it is critical modulation, and helps underpin the multi-billion dollar industry. Engineering is the design, analysis, and/or construction of works for practical purposes. ... Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). ... Mathematical physics is the scientific discipline concerned with the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories1. ... This article or section does not cite its references or sources. ... This article or section contains inappropriate citations. ... In mathematics and physics, a soliton is a self-reinforcing solitary wave caused by nonlinear effects in the medium. ... Fiber Optic strands An optical fiber in American English or fibre in British English is a transparent thin fiber for transmitting light. ... In telecommunications, modulation is the process of varying a periodic waveform, i. ...

Examples

Examples of solvable non-linear models include the KdV equation, the KP equation, the non-linear Schrödinger equation, the sine-Gordon equation, the Toda lattice. the Manakov model. Other important cases are discrete analogues of the famous Painlevé transcendental equations. The Korteweg-de Vries equation (KdV equation for short) is the following partial differential equation for a function φ of two real variables, x and t: Its solutions clump up into solitons. ... The Sine-Gordon equation is a partial differential equation for a function of two real variables, x and t, given as follows: The name is a pun on the Klein-Gordon equation. ...

Methods of solution

The inverse scattering method or more generally the use of Lax pairs is commonly applied to solve many of the models of this type. Although many models are integrable because they posses a symmetry, the constants of motion that result from the application of Noether's theorem are typically non-intuitive and non-obvious. In mathematics, in the theory of differential equations, a Lax pair is a pair of time-dependent matrices that describe certain solutions of differential equations. ... Sphere symmetry group o. ... In mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion. ... Noethers theorem is a central result in theoretical physics that shows that a conservation law can be derived from any continuous symmetry. ...

Such systems are an area of broad and deep mathematical research. Tools and techniques include the study of Hopf algebras, Poisson algebras and Poisson-Lie groups, since Poisson's theorem guarantees that the Poisson bracket of any two constants of motion is also a constant of motion. This line of study leads to the general area of quantum groups and non-commutative geometry. In mathematics, a Hopf algebra, named after Heinz Hopf, is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. ... A Poisson algebra is an associative algebra together with a Lie bracket, satisfying Leibniz law. ... In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. ... In abstract algebra, a Hopf algebra is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes . (Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. ... In mathematics, there is a close relationship between spaces, which are geometric in nature, and the numerical functions on them. ...

Some systems show a conformal symmetry, and thus have interesting relationships to the modular forms and Hecke algebras studied in number theory. In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a Euclidean-like space with a point added at infinity, or a Minkowski-like space with a couple of points added at infinity. That is, the setting is a compactification of a familiar space... A modular form is an analytic function on the upper half plane satisfying a certain kind of functional equation and growth condition. ... In mathematics, in particular in the theory of modular forms, a Hecke operator is a certain kind of averaging operator that plays a significant role in the structure of vector spaces of modular forms (and more general automorphic representations). ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ...

Some systems can be solved by means of supersymmetry, in which case the solution usually means the full low-energy effective action which includes the masses of BPS particles as functions of the moduli space. This article or section is in need of attention from an expert on the subject. ... Everything in the following article also applies to statistical mechanics. ... The Bogomolnyi-Prasad-Sommerfeld bound is a series of inequalities for solutions of partial differential equations depending on the homotopy class of the solution at infinity. ... In algebraic geometry, a moduli space is a parameter space for families of algebraic objects (such as algebraic varieties, morphisms, vector bundles). ...

Discrete systems

Parallel to partial differential equations and ordinary differential equations, there exist their discrete versions called partial difference equations and ordinary difference equations. In comparison with the analogous theories for differential equations, the development of rigorous analytic tools for difference equations is still in its infancy. In mathematics, and in particular analysis, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function. ... In mathematics, a differential equation is an equation that describes a prescribed relationship between a set of unknowns which are to be regarded as an unknown function and its (ordinary or partial) derivatives. ... In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ... In mathematics, a recurrence relation, also known as a difference equation, is an equation which defines a sequence recursively: each term of the sequence is defined as a function of the preceding terms. ...

Partial difference equations can also form integrable systems. For instance, the difference analogues of the soliton equations are integrable lattice equations. Not all discretization schemes are promising here. Application of brute force discretisation methods will typically destroy the key integrability properties (such as the existence of soliton solutions). To obtain genuine integrable discrete systems that preserve the key properties much subtler methods are needed. In mathematics and physics, a soliton is a self-reinforcing solitary wave caused by nonlinear effects in the medium. ... Discretization concerns the process of transferring continuous models and equations into discrete counterparts. ... Integrability is a mathematical concept used in different areas. ...

Quantum integrable models

In addition to classically integrable systems, there are quantum integrable models. The first solved quantum integrable model was the Heisenberg model, by Hans Bethe in 1931, using a method that has come to be known as the Bethe ansatz. These quantum models are almost exclusively one-dimensional. Formally equivalent to them are two-dimensional exactly solvable statistical mechanical models, such as the Ising model on a two-dimensional lattice. In both cases a key ingredient in solving the model is the Yang-Baxter equation. Classical mechanics is a branch of physics which studies the deterministic motion of objects. ... Fig. ... Hans Albrecht Bethe (pronounced bay-tuh; July 2, 1906 – March 6, 2005), was a German-American physicist who won the Nobel Prize in Physics in 1967 for his work on the theory of stellar nucleosynthesis. ... In physics, the Bethe Ansatz is a method for finding the exact solutions of certain quantum many-body models. ... The Ising model, named after the physicist Ernst Ising, is a mathematical model in statistical mechanics. ... The Yang-Baxter equation is an equation which was first introduced in the field of statistical mechanics. ...