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On the integrability of the inhomogeneous spherically symmetric Heisenberg ferromagnet in arbitrary dimensions. (English) Zbl 0821.35114

Summary: 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 \(f(r)= \varepsilon_ 1 r^{-2(n -1)}+ \varepsilon_ 2 r^{-(n- 2)}\). This is confirmed by obtaining the associated Lax pair, Bäcklund transformation, and the soliton like 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.

MSC:

35Q40 PDEs in connection with quantum mechanics
81T25 Quantum field theory on lattices
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q55 NLS equations (nonlinear Schrödinger equations)
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