Exact and non-exact Fermi-Pasta-Ulam-Tsingou recurrences in a Heisenberg ferromagnet

We visualize the Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence in a classical Heisenberg ferromagnetic (HF) spin chain by exploiting its gauge eq uivalence to the nonlinear Schr\"{o}dinger equation (NLSE). We discuss two types of spatially periodic breather excitations in the spin chain, that are associated with: (I) Akhmediev breather, and (II) Galilean transformed Akhmediev breather. The recurrence in the former is exact in the sense that the initial and final states are identical. In the later, the spin chain undergoes an additional global rotation during the rec urrence process, which makes the initial and final states distinguishable. Both the complex solutions (I) and (II) nevertheless show a definit e phase shift during the recurrence process. A one-to-one correspondence between HF spin chain and the NLSE seems missing by virtue of the clo seness of the FPUT recurrence.