The dynamics of a generalized Heisenberg ferromagnetic spin chain

We introduce a generalized Heisenberg ferromagnetic spin chain with a four-dimensional target space, and investigate its continuous limit, the generalized continuous Heisenberg model (GCHM). We reduce the dynamics of the GCHM to a nonlinear evolution of space curves in four dimensions. The space curve evolution is expressed in terms of a system of coupled nonlinear equations for the three curvatures, k(1)(x),k(2)(x),k(3)(x), of a curve in R(4). Applying the Painleve analysis to the stationary equations, we conclude that GCHM, in general, is not integrable, unless k(1) is constant. We obtain explicit solutions of the resulting stationary system under the latter condition. (c) 1995 American Institute of Physics.