Dispersion law for a one-dimensional weakly interacting Bose gas with zero boundary conditions

From the time-dependent Gross equation, we find the quasiparticle dispersion law for a one-dimensional weakly interacting Bose gas with a non-point interatomic potential and zero boundary conditions (BCs). The result coincides with the dispersion law for periodic BCs, i.e. the Bogolyubov law $E_{B}(k) = \sqrt{\left (\frac{\hbar^{2} k^2}{2m}\right )^{2} + n_{0}\nu(k)\frac{\hbar^2 k^2}{m}}$. In the case of periodic BCs, the dispersion law can be easily derived from Gross' equation. However, for zero BCs, the analysis is not so simple.