Quantum dark solitons in a Bose gas confined in a hard-wall box

The Schr\"odinger equation for a Bose gas with repulsive contact interactions in a one-dimensional space may be solved analytically with the help of the Bethe ansatz if we impose periodic boundary conditions. It was shown that for such a system there exist many-body eigenstates directly corresponding to dark soliton solutions of the mean-field equation. The system is still integrable if one switches from the periodic boundary conditions to an infinite-square-well potential. The corresponding eigenstates were constructed by Gaudin. We analyze the weak-interaction limit of Gaudin's solutions and identify the parametrization of eigenstates strictly connected with single and multiple dark solitons. Numerical simulations of measurements of the particles' positions reveal dark solitons in the weak-interaction regime and their quantum nature in the presence of strong interactions.