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We investigate the dynamics of the one-dimensional strongly repulsivespin-1/2 Bose-Hubbard model for filling $\nu\le1.$ While at $\nu=1$ the systemis a Hubbard-Mott insulator exhibiting dynamical properties of the Heisenbergferromagnet, at $\nu<1$ it is a ferromagnetic liquid with complex spindynamics. We find that close to the insulator-liquid transition the systemadmits for a complete separation of spin and density degrees of freedom validat {\it all} energy and momentum scales within the $t-J$ approximation. Thisallows us to derive the propagator of transverse spin waves and the shape ofthe magnon peak in the dynamic spin structure factor.

Dynamical properties of the one-dimensional spin-1/2 Bose-Hubbard model near Mott-insulator to ferromagnetic liquid transition