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 is 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, 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 statistics, 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 share... 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) equation 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 application of scientific and technical knowledge to solve human problems. ... Numerical analysis is the study of algorithms 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. ... Buyers bargain for good prices while sellers put forth their best front in Chichicastenango Market, Guatemala. ... 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. ... Modulation is the process of varying a carrier signal, typically a sinusoidal signal, in order to use that signal to convey information. ...

Examples

Examples of solvable non-linear models include the KdV equation, the KP equation, the non-linear Schrödinger equation, the sine-Gordon equation, the Manakov model and the Yang-Baxter equation. 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. ... The Yang-Baxter equation is an equation which was first introduced in the field of statistical mechanics. ...

Examples of discrete systems or lattices, some variants of which are solvable, are the Ising model, the Potts model, and the Toda lattice. Other important cases are discrete analogues of the famous Painlevé transcendental equations. The Ising model, named after the physicist Ernst Ising, is a mathematical model in statistical mechanics. ... In statistical mechanics, the Potts model, a generalization of the Ising model, is a model of interacting spins on a crystalline lattice. ...

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. Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... Noethers theorem is a central result in theoretical physics that expresses the one-to-one correspondence between symmetries and conservation laws. ...

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 gaurantees 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 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. ... 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). ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

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. ...