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The thesis contains description of the Lieb-Liniger model in the context ofthe correspondence between dark solitons and the so-called hole excitations. Wepresent a detailed analysis of the analytical solution given by the Betheansatz and discuss two types of elementary - particle (type I) and hole (typeII) - excitations. It turns out that the type I excitations are reproduced bythe Bogoliubov spectrum which means that they correspond to the sound waves inthe system. It is believed that the eigenstates corresponding to the secondbranch are strictly connected with dark solitons. The main evidence bases onthe comparison between the spectrum of the hole excitations and the dispersionrelation of the semi-classical soliton. All the knowledge needed to fullyunderstand the problem is presented in details in the thesis. Our numericalsimulations show that successive measurement of particle positions is able tobreak the translation symmetry of the system and reveal particle densitiesexpected from the dark soliton profiles if the system is prepared in a type IIeigenstate. We analyze single and double dark solitons in weak and stronginteraction regime.

Emergence of dark solitons in the course of measurements of particle positions in the Lieb-Liniger model: detailed analysis