Integrable system
In mathematics, integrability is a property of certain
Three features are often referred to as characterizing integrable systems:[1]
- the existence of a maximal set of conserved quantities (the usual defining property of complete integrability)
- the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic integrability)
- the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability)
Integrable systems may be seen as very different in qualitative character from more generic dynamical systems, which are more typically chaotic systems. The latter generally have no conserved quantities, and are asymptotically intractable, since an arbitrarily small perturbation in initial conditions may lead to arbitrarily large deviations in their trajectories over a sufficiently large time.
Many systems studied in physics are completely integrable, in particular, in the
In the late 1960s, it was realized that there are completely integrable systems in physics having an infinite number of degrees of freedom, such as some models of shallow water waves (
In the special case of Hamiltonian systems, if there are enough independent Poisson commuting first integrals for the flow parameters to be able to serve as a coordinate system on the invariant level sets (the leaves of the
A key ingredient in characterizing integrable systems is the
Integrability does not necessarily imply that generic solutions can be explicitly expressed in terms of some known set of special functions; it is an intrinsic property of the geometry and topology of the system, and the nature of the dynamics.
General dynamical systems
In the context of differentiable
An extension of the notion of integrability is also applicable to discrete systems such as lattices. This definition can be adapted to describe evolution equations that either are systems of
The distinction between integrable and nonintegrable dynamical systems has the qualitative implication of regular motion vs.
Hamiltonian systems and Liouville integrability
In the special setting of
In finite dimensions, if the phase space is symplectic (i.e., the center of the Poisson algebra consists only of constants), it must have even dimension and the maximal number of independent Poisson commuting invariants (including the Hamiltonian itself) is . The leaves of the foliation are
There is also a distinction between complete integrability, in the
Action-angle variables
When a finite-dimensional Hamiltonian system is completely integrable in the Liouville sense, and the energy level sets are compact, the flows are complete, and the leaves of the invariant foliation are
The Hamilton–Jacobi approach
In
Solitons and inverse spectral methods
A resurgence of interest in classical integrable systems came with the discovery, in the late 1960s, that
The basic idea of this method is to introduce a linear operator that is determined by the position in phase space and which evolves under the dynamics of the system in question in such a way that its "spectrum" (in a suitably generalized sense) is invariant under the evolution, cf. Lax pair. This provides, in certain cases, enough invariants, or "integrals of motion" to make the system completely integrable. In the case of systems having an infinite number of degrees of freedom, such as the KdV equation, this is not sufficient to make precise the property of Liouville integrability. However, for suitably defined boundary conditions, the spectral transform can, in fact, be interpreted as a transformation to completely ignorable coordinates, in which the conserved quantities form half of a doubly infinite set of canonical coordinates, and the flow linearizes in these. In some cases, this may even be seen as a transformation to action-angle variables, although typically only a finite number of the "position" variables are actually angle coordinates, and the rest are noncompact.
Hirota bilinear equations and τ-functions
Another viewpoint that arose in the modern theory of integrable systems originated in a calculational approach pioneered by Ryogo Hirota,[2] which involved replacing the original nonlinear dynamical system with a bilinear system of constant coefficient equations for an auxiliary quantity, which later came to be known as the τ-function. These are now referred to as the Hirota equations. Although originally appearing just as a calculational device, without any clear relation to the
Subsequently, this was interpreted by Mikio Sato[3] and his students,[4][5] at first for the case of integrable hierarchies of PDEs, such as the Kadomtsev–Petviashvili hierarchy, but then for much more general classes of integrable hierarchies, as a sort of universal phase space approach, in which, typically, the commuting dynamics were viewed simply as determined by a fixed (finite or infinite) abelian group action on a (finite or infinite) Grassmann manifold. The τ-function was viewed as the determinant of a
Quantum integrable systems
There is also a notion of quantum integrable systems.
In the quantum setting, functions on phase space must be replaced by
To explain quantum integrability, it is helpful to consider the free particle setting. Here all dynamics are one-body reducible. A quantum system is said to be integrable if the dynamics are two-body reducible. The
Exactly solvable models
In physics, completely integrable systems, especially in the infinite-dimensional setting, are often referred to as exactly solvable models. This obscures the distinction between integrability, in the Hamiltonian sense, and the more general dynamical systems sense.
There are also exactly solvable models in statistical mechanics, which are more closely related to quantum integrable systems than classical ones. Two closely related methods: the Bethe ansatz approach, in its modern sense, based on the Yang–Baxter equations and the quantum inverse scattering method, provide quantum analogs of the inverse spectral methods. These are equally important in the study of solvable models in statistical mechanics.
An imprecise notion of "exact solvability" as meaning: "The solutions can be expressed explicitly in terms of some previously known functions" is also sometimes used, as though this were an intrinsic property of the system itself, rather than the purely calculational feature that we happen to have some "known" functions available, in terms of which the solutions may be expressed. This notion has no intrinsic meaning, since what is meant by "known" functions very often is defined precisely by the fact that they satisfy certain given equations, and the list of such "known functions" is constantly growing. Although such a characterization of "integrability" has no intrinsic validity, it often implies the sort of regularity that is to be expected in integrable systems. [citation needed]
List of some well-known integrable systems
- Classical mechanical systems
- Calogero–Moser–Sutherland model[9]
- Central force motion (exact solutions of classical central-force problems)
- Geodesic motion on ellipsoids
- Harmonic oscillator
- Integrable Clebsch and Steklov systems in fluids
- Lagrange, Euler, and Kovalevskaya tops
- Neumann oscillator
- Two center Newtonian gravitationalmotion
- Integrable lattice models
- Ablowitz–Ladik lattice
- Toda lattice
- Volterra lattice
- Integrable systems in 1 + 1 dimensions
- AKNS system
- Benjamin–Ono equation
- Boussinesq equation (water waves)
- Camassa–Holm equation
- Classical Heisenberg ferromagnet model (spin chain)
- Degasperis–Procesi equation
- Dym equation
- Garnier integrable system
- Kaup–Kupershmidt equation
- Krichever–Novikov equation
- Korteweg–de Vries equation
- Landau–Lifshitz equation (continuous spin field)
- Nonlinear Schrödinger equation
- Nonlinear sigma models
- Sine–Gordon equation
- Thirring model
- Three-wave equation
- Integrable PDEs in 2 + 1 dimensions
- Davey–Stewartson equation
- Ishimori equation
- Kadomtsev–Petviashvili equation
- Novikov–Veselov equation
- Integrable PDEs in 3 + 1 dimensions
- The Einstein field equations; general solutions are termed gravitational solitons, of which the Schwarzschild metric, the Kerr metric and some gravitational wavesolutions are examples.
- Exactly solvable statistical lattice models
- 8-vertex model
- Gaudin model
- Ising model in 1- and 2-dimensions
- Ice-type model of Lieb
- Quantum Heisenberg model
See also
Related areas
- Mathematical physics
- Soliton
- Painleve transcendents
- Statistical mechanics
- Integrable algorithm
Some key contributors (since 1965)
- Mark Ablowitz
- Rodney Baxter
- Percy Deift
- Leonid Dickey
- Vladimir Drinfeld
- Boris Dubrovin
- Ludvig Faddeev
- Hermann Flaschka
- Israel Gel'fand
- Alexander Its
- Michio Jimbo
- Igor M. Krichever
- Martin Kruskal
- Peter Lax
- Vladimir Matveev
- Henry McKean
- Robert Miura
- Tetsuji Miwa
- Alan Newell
- Nicolai Reshetikhin
- Aleksei Shabat
- Evgeny Sklyanin
- Mikio Sato
- Elliott H. Lieb
- Graeme Segal
- George Wilson
- Vladimir E. Zakharov
References
- ISBN 978-0-387-96890-2.
- ISBN 978-0521779197.
- Babelon, O.; Bernard, D.; Talon, M. (2003). Introduction to classical integrable systems. ISBN 0-521-82267-X.
- ISBN 978-0-12-083180-7.
- Dunajski, M. (2009). Solitons, Instantons and Twistors. ISBN 978-0-19-857063-9.
- ISBN 978-0-387-15579-1.
- ISBN 978-2-88124-901-3.
- ISBN 978-0-415-29805-6.
- ISBN 0-201-02918-9.
- ISBN 0-8218-2093-1.
- S2CID 222379146.
- Hietarinta, J.; ISBN 978-1-107-04272-8.
- ISBN 978-0-521-58646-7.
- ISBN 3-540-18173-3.
- Mussardo, Giuseppe (2010). Statistical Field Theory. An Introduction to Exactly Solved Models of Statistical Physics. ISBN 978-0-19-954758-6.
- ISBN 978-5-396-00687-4.
Further reading
- Beilinson, A.; Drinfeld, V. "Quantization of Hitchin's integrable system and Hecke eigensheaves" (PDF).
- ISBN 978-3-540-60542-3.
- Sonnad, Kiran G.; Cary, John R. (2004). "Finding a nonlinear lattice with improved integrability using Lie transform perturbation theory". Physical Review E. 69 (5): 056501. PMID 15244955.
External links
- "Integrable system", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- "SIDE - Symmetries and Integrability of Difference Equations", a conference devoted to the study of integrable difference equations and related topics.[10]
Notes
- ISBN 978-0-19-967677-4.
- .
- hdl:2433/102800.
- .
- .
- S2CID 124170507.
- ISBN 978-0-521-58646-7.
- S2CID 119331736.
- .
- ISBN 978-0-521-59699-2.