integrablesystems

Integrable systems is a broad term covering dynamical systems that can be solved exactly because of a rich underlying structure, typically a large number of conserved quantities or a special algebraic formulation. In classical mechanics, a Hamiltonian system with N degrees of freedom is Liouville integrable if it possesses N independent first integrals that are in involution with respect to the Poisson bracket. When this condition holds, the motion can be described by action-angle variables and reduces to linear motion on an N-dimensional torus, yielding regular, nonchaotic dynamics.

In the study of nonlinear partial differential equations, integrable systems include many famous evolution equations such as the Korteweg–de Vries (KdV), nonlinear Schrödinger (NLS), sine-Gordon, and Kadomtsev–Petviashvili (KP) equations. These PDEs admit an infinite hierarchy of conserved quantities, exact multi-soliton solutions, and powerful solution methods. A central concept is the Lax pair, a pair of linear operators whose compatibility condition reproduces the nonlinear evolution; from a Lax representation one can apply the inverse scattering transform, spectral methods, or algebro-geometric techniques to construct explicit solutions. Discrete and lattice versions, such as the Toda lattice or discrete KdV, are also integrable and share many of these features.

Quantum integrable systems extend the idea to quantum mechanics, where there exists a commuting family of quantum operators. Methods such as the Bethe Ansatz and the Yang–Baxter equation yield exact spectra for models like the Heisenberg spin chain and the one-dimensional Bose gas. In all settings, integrable systems stand out for their exact solvability, stable structures, and rich mathematical connections, though most nonlinear systems are not integrable.