Integrable Hierarchy

An integrable hierarchy is an infinite family of mutually compatible evolution equations, usually represented by commuting flows in independent time variables , , .... A single nonlinear partial differential equation can occur as one flow or as a reduction of a hierarchy.

Many integrable hierarchies have a Lax pair formulation in which an operator evolves according to

Here the brackets denote the commutator. Compatibility of two flows is expressed by the zero-curvature equation

The Kadomtsev-Petviashvili equation occurs in the KP hierarchy, and reductions of that hierarchy include the Korteweg-de Vries equation hierarchy. Such hierarchies are associated with infinitely many commuting symmetries and conserved quantities.