  Using the algebro-geometric approach, we study the structure of
semi-classical eigenstates in a weakly-anisotropic quantum Heisenberg spin
chain. We outline how classical nonlinear spin waves governed by the
anisotropic Landau-Lifshitz equation arise as coherent macroscopic low-energy
fluctuations of the ferromagnetic ground state. Special emphasis is devoted to
the simplest types of solutions, describing precessional motion and elliptic
magnetisation waves. The internal magnon structure of classical spin waves is
resolved by performing the semi-classical quantisation using the
Riemann-Hilbert problem approach. We present an expression for the overlap of
two semi-classical eigenstates and discuss how correlation functions at the
semi-classical level arise from classical phase-space averaging.
