On the integrability of the inhomogeneous spherically symmetric Heisenberg ferromagnet in arbitrary dimensions

Abstract

The dynamics of an inhomogeneous spherically symmetric continuum Heisenberg ferromagnet in arbitrary (n-) dimensions is considered. By a known geometrical procedure the spin evolution equation equivalently is rewritten as a generalized nonlinear Schrödinger equation. A Painlevé singularity structure analysis of the solutions of the equation shows that the system is integrable in arbitrary (n-) dimensions only when the inhomogeneity is of inverse power in the radial coordinate in the form f(r)=ε1r−2(n−1)+ε2r−(n−2). This is confirmed by obtaining the associated Lax pair, Bäcklund transformation, and the solitonlike solution of the evolution equation. Further, calculations show that the one-dimensional linearly inhomogeneous ferromagnet acts as a universal model to which all the integrable higher-dimensional inhomogeneous spherically symmetric spin models can be formally mapped.