  We study the relaxation of excitations in a system of one-dimensional weakly
interacting bosons. Due to residual weak interactions, Bogoliubov
quasiparticles in this system have finite lifetimes. As a result of the
conservation laws in one dimension, at zero temperature the leading mechanism
of decay of a quasiparticle is disintegration into three others. We focus on
phonon quasiparticles and find that their decay rate is proportional to the
seventh power of momentum. In the integrable case of contact interaction
between the bosons, the decay rate vanishes.
