Surface-embeddability approach to the dynamics of the inhomogeneous Heisenberg spin chain

Abstract

The surface-embeddability approach of Lund and Regge is applied to the classical, inhomogeneous Heisenberg spin chain to study the class of inhomogeneity functions f for which the spin evolution equation and its gauge-equivalent generalized nonlinear Schrödinger equation (GNLSE) are exactly solvable. Writing the spin vector S(x,t) as ∂xr and identifying r(x,t) with a position vector generating a surface, we show that the kinematic equation satisfied by r implies certain constraints on the admissible geometries of this surface. These constraints, together with the Gauss–Mainardi–Codazzi equations, enable us to express the coefficient of the second fundamental form as well as f in terms of the metric coefficients G and its derivatives, for arbitrary time-independent G. Explicit solutions for the GNLSE can also be found in terms of the same quantities. Of the admissible surfaces generated by r, a special class that emerges naturally is that of surfaces of revolution: Explicit solutions for r and S are found and discussed for this class of surfaces.