Exactly solvable model
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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.
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.
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.
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.
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[edit] 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.
[edit] 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.
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.
Some systems show a conformal symmetry, and thus have interesting relationships to the modular forms and Hecke algebras studied in number theory.
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.
[edit] 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.
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.
[edit] 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.
