Nonlocal magnon density and breather formation in one-dimensional ferromagnets

We investigate a nonlocal generalization of the nonlinear Schrödinger equation describing magnetic solitons in a one-dimensional ferromagnet embedded in a spin-wave background. In this setting, the wave field at a given position is coupled to its spatially mirrored counterpart, introducing a nonlocal interaction that qualitatively alters soliton dynamics relative to the standard local model. Using the Darboux transformation, we obtain exact analytical solutions and reconstruct the magnetization evolution. Our results have shown that the effective magnon density is influenced by nonlocal correlations instead of the local field intensity only. When the spin-wave background is absent, the soliton is immobile, which is in agreement with the classical scenario. In contrast, if there is a finite background, the soliton will decay into a breather, which has periodic oscillations of amplitude as it exchanges magnons with the medium. The underlying nonlocal symmetry locks the soliton at the origin of space and plays a central role in determining its long time behaviour causing such effects as recurrent emission of spin wave packets and, in the case of strong symmetry breaking, the onset of finite time blow up. These analytical predictions are supported by numerical simulations which lead to a direct comparison of local and nonlocal dynamics with a special focus on the respiratory emergence of breathing solitons together with their possible relevance for the manipulation of spin-wave processes in magnetic systems.