  We reveal the intricate impact of nonlinearity and disorder on the thermal
conductivity of acoustic chains. Disorder induces mobility edges and allows to
control the amount of extended modes which are the ballistic channels for
energy transport. Nonlinearity adds a diffusive conductivity channel through
interacting localized modes, and controls the contact resistance at the edges.
Analytical arguments and numerical results yield several crossovers between
dominating heat channels, when varying the system size and the temperature. We
demonstrate that the nonlinearity-induced interaction between the modes alters
the transport through existing conductivity channels and creates new ones,
underpinning the observed phenomena.
