Violation of Self-Duality for Topological Solitons due to Soliton-Soliton Interaction on a Cylindrical Geometry

We study classical Heisenberg spins on an infinite elastic cylinder. In the continuum limit the Hamiltonian of the system is given by the nonlinear σ model. We investigate the periodic, cylindrically symmetric solution of the sine-Gordon equation (the Euler-Lagrange equation for this Hamiltonian). The solution does not satisfy the self-dual equations of Bogomol'nyi [Sov. J. Nucl. Phys. 24, 449 (1976)] which give the minimum energy configuration in each homotopy class. This leads to a novel geometric effect: periodic shrinking of the cylinder