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

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