Nonlinear Schrödinger equations and simple Lie algebras

We associate a system of integrable, generalised nonlinear Schrödinger (NLS) equations with each Hermitian symmetric space. These NLS equations are considered as reductions of more general systems, this time associated with a reductive homogeneous space. The nonlinear terms are related to the curvature and torsion tensors of the appropriate geometrical space. The Hamiltonian structure is investigated using “r-matrix” techniques and shown to be “canonical” for all these equations. Throughout the reduction procedure this Hamiltonian structure does not degenerate. Each of the above systems of equations is gauge equivalent to a generalised ferromagnet. Reductions of the latter are discussed in terms of the corresponding NLS type equations.