  The thesis contains description of the Lieb-Liniger model in the context of
the correspondence between dark solitons and the so-called hole excitations. We
present a detailed analysis of the analytical solution given by the Bethe
ansatz and discuss two types of elementary - particle (type I) and hole (type
II) - excitations. It turns out that the type I excitations are reproduced by
the Bogoliubov spectrum which means that they correspond to the sound waves in
the system. It is believed that the eigenstates corresponding to the second
branch are strictly connected with dark solitons. The main evidence bases on
the comparison between the spectrum of the hole excitations and the dispersion
relation of the semi-classical soliton. All the knowledge needed to fully
understand the problem is presented in details in the thesis. Our numerical
simulations show that successive measurement of particle positions is able to
break the translation symmetry of the system and reveal particle densities
expected from the dark soliton profiles if the system is prepared in a type II
eigenstate. We analyze single and double dark solitons in weak and strong
interaction regime.
