The n × n KdV hierarchy

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${sl (n, \mathbb{C})}$$\end{document} . The flows in the hierarchy include systems of coupled nonlinear Schrödinger equations. Since they are constructed from a Lie algebra splitting, the general method gives formal inverse scattering, bi-Hamiltonian structures, commuting flows, and Bäcklund transformations for these hierarchies.