Symmetries and Integrability

Abstract

Integrable nonlinear evolution equations in one‐spatial and one‐temporal dimensions possess a remarkably rich algebraic structure: Infinitely many symmetries and conserved quantities, existence of a bi‐Hamiltonian formulation etc. A certain operator Φ, called recursion operator, plays a central role in investigating the above algebraic properties. Recently the above theory has been extended to equations in two spatial and one temporal dimensions. In particular, the multidimensional analogue of the operator Φ has been found and its bi‐Hamiltonian factorization has been established. The above and other aspects of integrable equations, including mastersymmetries, are reviewed in this manuscript. Furthermore, a definition of integrability is proposed based on symmetry considerations.