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.