Classical Spin Systems, Nonlinear Evolution Equations and Nonlinear Excitations

Abstract

Nonlinear excitations in classical generalized Heisenberg spin systems are studied with particular attention paid to domain-wall and solitary-wave excitations. Two kinds of nonlinear evolution equations obeyed either by the conventional two angles of rotation or the stereographic variables are derived in the form of nonlinear differential-difference equations for the isotropic and anisotropic Heisenberg models and the XY model, each of which is in an external magnetic field. It is shown by employing a continuum approximation that nonlinear excitations in classical spin systems described by various kinds of nonlinear differential equations are very rich in the sense that different spin model and also different spatial dimensionality yield different type of soliton-like excitations. The nonlinear differential equations written in terms of the stereographic variables have stronger nonlinearity than those appearing in other fields. A detailed study of static solutions of the equations is made in one-dimensional case for the anisotropic Heisenberg model and the XY model. It is shown that a variety of domain-wall solutions can exist, one is associated with symmetry-breaking states and the others are 2π -and π-domain-wall solutions. Brief discussions are also given to stationary solutions of the nonlinear differential equations in two- and three-dimensional cases to elucidate the existence of different nature of nonlinear excitations.