  An influence of the phonon-phonon coupling (PPC) on the $\beta$-decay
half-lives and multi-neutron emission probabilities is analysed within the
microscopic model based on the Skyrme interaction with tensor components
included. The finite-rank separable approximation is used in order to handle
large two-quasiparticle spaces. The even-even nuclei near the r-process pathes
at $N=82$ are studied. The characteristics of ground states, $2^+$ excitations
and $\beta$-decay strength of the neutron-rich Cd isotopes are treated in
detail. It is shown that a strong redistribution of the Gamow-Teller strength
due to the PPC is mostly sensitive to the multi-neutron emission probability of
the Cd isotopes.
