In this talk, we shall consider the classical field theory of the Heisenberg ferromagnet, which is the semi-classical limit of the Heisenberg spin-1/2 chain. Our primary motivation to study classical solvable dynamical system is to understand the similarities and differences between integrable quantum systems and their classical counterparts. Unfortunately, there seem to exist no efficient computational framework to achieve this goal. To this end, by starting from first principles and employing the finite-gap algebro-geometric integration technique, we identify a certain scaling procedure of the corresponding degenerate large-genus Riemann surfaces which provides a statistical description for a thermodynamic gas of magnetic solitons. Using simple arguments of semi-classical quantization, we use the classical S-matrix to derive a universal integral dressing equation for the spectral distribution function of soliton excitations which accounts for interactions with a non-trivial many-body vacuum. By lifting the conventional theory of Whitham modulations equations to the thermodynamic setting, we obtain a simple formula for the effective propagation velocity which is, remarkably, in formal agreement with that proposed recently for quantum integrable models.

* Enej Ilievski, Universiteit van Amsterdam