  We theoretically investigate the effects of atom losses in the
one-dimensional (1D) Bose gas with repulsive contact interactions, a famous
quantum integrable system also known as the Lieb-Liniger gas. The generic case
of K-body losses (K = 1,2,3,...) is considered. We assume that the loss rate is
much smaller than the rate of intrinsic relaxation of the system, so that at
any time the state of the system is captured by its rapidity distribution (or,
equivalently, by a Generalized Gibbs Ensemble). We give the equation governing
the time evolution of the rapidity distribution and we propose a general
numerical procedure to solve it. In the asymptotic regimes of vanishing
repulsion -- where the gas behaves like an ideal Bose gas -- and hard-core
repulsion -- where the gas is mapped to a non-interacting Fermi gas -- we
derive analytic formulas. In the latter case, our analytic result shows that
losses affect the rapidity distribution in a non-trivial way, the time
derivative of the rapidity distribution being both non-linear and non-local in
rapidity space.
