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We investigate field-induced transformations in the dynamical response of the$XXZ$ model on the triangular lattice that are associated with the anharmonicmagnon coupling and decay phenomena. A set of concrete theoretical predictionsis made for a close physical realization of the spin-1/2 $XXZ$ model,Ba$_3$CoSb$_2$O$_9$. We demonstrate that dramatic modifications in magnonspectrum must occur in low out-of-plane fields that are easily achievable forthis material. The hallmark of the effect is a coexistence of the clearlydistinct well-defined magnon excitations with significantly broadened ones indifferent regions of the ${\bf k}-\omega$ space. The field-induced decays aregeneric for this class of models and become more prominent at largeranisotropies and in higher fields.

Field-induced decays in $XXZ$ triangular-lattice antiferromagnets