The n × n KdV hierarchy

Abstract

We introduce two new soliton hierarchies that are generalizations of the KdV hierarchy. Our hierarchies are restrictions of the AKNS n × n hierarchy coming from two unusual splittings of the loop algebra. These splittings come from automorphisms of the loop algebra instead of automorphisms of \({sl (n, \mathbb{C})}\) . The flows in the hierarchy include systems of coupled nonlinear Schrodinger equations. Since they are constructed from a Lie algebra splitting, the general method gives formal inverse scattering, bi-Hamiltonian structures, commuting flows, and Backlund transformations for these hierarchies.