On the evolution of higher dimensional Heisenberg continuum spin systems

We consider the evolution of a classical Heisenberg ferromagnetic spin chain in its continuum limit in higher spatial dimensions. It is shown that the evolution of a radially symmetric chain could be identified with the motion of a helical space curve as in the linear case. The resulting invariant equations for the curvature (radial energy density) and torsion (related to current density) are shown to be equivalent to a generalized nonlinear Schrödinger equation, similar to the one derived by Ruijgrok and Jurkiewicz recently. Equivalent linear equations as well as special static solutions of point singular type are obtained. Similarity solutions, a class of which belong to Riccati type, are discussed in detail. For general higher dimensions, a potentially useful formulation is presented: Under stereographic projection of the unit sphere of spin, the equation of motion takes a neater form even with the inclusion of anisotropic interactions. Classes of explicit solutions are reported in higher dimensions. Propagating spin waves, static spin waves of point singular nature and of finite energy in some cases are also discussed.