  We study the exact solutions for a one-dimensional system of $N=2; 3$
spinless point bosons for zero boundary conditions. In this case, we are based
on M. Gaudin's formulae obtained with the help of Bethe ansatz. We find the
density profile $\rho(x)$ and the nodal structure of a wave function for a set
of the lowest states of the system for different values of the coupling
constant $\gamma\geq 0$. The analysis shows that the ideal crystal corresponds
to the quantum numbers (from Gaudin's equations) $n_{1}=\ldots =n_{N}=N$ and to
the coupling constant $\gamma \leq 1$. We also find that the ground state of
the system ($n_{1}=\ldots =n_{N}=1$) corresponds to a liquid for any $\gamma$
and any $N\gg 1$. In this case, the wave function of the ground state is
nodeless, and the wave function of the ideal crystal has nodes.
