Rogue wave dynamics and energy spectra for the generalized Heisenberg spin chain equation

We conduct a systematic investigation on the rogue wave structures of the generalized Heisenberg spin chain equation, offering potential insights into the dynamics of nonlinear waves in some ferromagnetic materials. Firstly, an iterative generalized Darboux transformation is proposed to obtain the exact high-order rogue waves on the plane wave backgrounds. Particularly, it is interesting that we also successfully derive the energy spectra of rogue waves via the fast Fourier transform routine, which reveals significant difference compared with the triangular spectra of the nonlinear Schr & ouml;dinger equation. Secondly, we extend the representation of large parameter asymptotic analysis to any order rogue wave solutions of the generalized Heisenberg spin chain equation for the first time. Depending on varying parameters, rogue waves display distinct separation patterns, encompassing triangles, pentagons, heptagons and even pentagrams. Finally, we numerically simulate the dynamical behaviors of some rogue waves via the split-step Fourier transform method. These results may be valuable to the exploration of physical importance for Heisenberg-type system in ferromagnet.