  Low-frequency modes of excitation in deformed neutron-rich nuclei are studied
by means of the quasiparticle random-phase approximation on the
Skyrme-Hartree-Fock-Bogoliubov mean field. We investigate the microscopic
structure of the soft $K^{\pi}=0^{+}$ modes systematically in neutron-rich
Magnesium isotopes with $N=22, 24, 26$ and 28 close to the drip line, and it is
found that the strong collectivity in $^{34}$Mg and $^{40}$Mg is acquired due
to the coherent coupling between the $\beta$ vibration and the pairing
vibration of neutrons. Microscopic structure of the $K^{\pi}=2^{+}$ modes
changes gradually associated with the location of the Fermi level of neutrons,
and it is found that the proton particle-hole excitation generating the
$\gamma-$vibrational mode in $^{24}$Mg continues to play a key role in the
near-drip-line nucleus $^{40}$Mg. The low-frequency octupole excitations are
also investigated and the microscopic mechanism for the enhancement of
transition strengths is discussed.
