A FRACTIONAL KdV HIERARCHY

Abstract

We construct a new system of integrable nonlinear differential equations associated with. the operator algebra [Formula: see text] of Polyakov. Its members are fractional generalizations of KdV type flows corresponding to an alternative set of constraints on the 2-dim. SL(3) gauge connections. We obtain the first non-trivial examples by dimensional reductiion from self-dual Yang–Mills and then generate recursively the entire hierarchy and its conserved quantities using a bi-Hamiltonian structure. Certain relations with the Boussinesq equation are also discussed together with possible generalizations of the formalism to SL (N) gauge groups and [Formula: see text] operator algebras with arbitrary N and l.