In the present work we study the transport properties of the lattice Landau-Lifshitz model in one spatial dimensions, which represents a classical analogue of a quantum spin chain. The model is motivated by a quantum mechanical treatment of the
exchange interaction. We introduce the concept of integrability in classical mechanics and write down the model’s equations of motion using Lax operators and the zero-curvature condition. The integrability of the models follows from an involutive property of a pair of monodromy matrices which we prove using an R-matrix. We present the Trotter decomposition of the Liouville equation, which allows for an efficient simulation of the model dynamics. The naive decomposition of the model turns out to be nonintegrable, which we verify by computing the Lyapunov spec-
trum. We introduce a novel integrable generalization of the Landau-Lifshitz model on the discrete time lattice and compute the two-body Hamiltonian that generates the requisite dynamics and find a novel R-matrix. We show that the model is self-dual. The model turns out to be superdiffusive in a non-magnetized state, by analyzing its correlation functions we further show it belongs into the Kardar-Parisi-Zhang universality class. In a magnetized state ballistic trasnport predominates, a rich dependance of the correlation function of magnetization upon net magnetization is observed.
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