  We study the finite temperature dynamical structure factor $S(k,\omega)$ of a
1D Bose gas using numerical simulations of the Gross--Pitaevskii equation
appropriate to a weakly interacting system. The lineshape of the phonon peaks
in $S(k,\omega)$ has a width $\propto |k|^{3/2}$ at low wavevectors. This
anomalous width arises from resonant three-phonon interactions, and reveals a
remarkable connection to the Kardar--Parisi--Zhang universality class of
dynamical critical phenomena.
