  We study two popular one-dimensional chains of classical anharmonic
oscillators: the rotor chain and a version of the discrete non-linear
Schr\"odinger chain. We assume that the interaction between neighboring
oscillators, controlled by the parameter $\epsilon >0$, is small. We rigorously
establish that the thermal conductivity of the chains has a non-perturbative
origin, with respect to the coupling constant $\epsilon$, and we provide strong
evidence that it decays faster than any power law in $\epsilon$ as $\epsilon
\rightarrow 0$. The weak coupling regime also translates into a high
temperature regime, suggesting that the conductivity vanishes faster than any
power of the inverse temperature.
