Multicomponent Nonlinear Evolution Equations of the Heisenberg Ferromagnet Type. Local versus Nonlocal Reductions

Abstract: The paper is dedicated to a system of matrix nonlinear evolution equations related to a Hermitian symmetric space of the type $\mathbf{A.III}$. The system under consideration extends the $1+1$ dimensional Heisenberg ferromagnet equation in the sense that its Lax pair has a form rather similar to the pair of the original Heisenberg ferromagnet model. We shall present here certain local and nonlocal reductions. A local integrable deformation and some of its reductions will be discussed too.