Exact solutions for the dispersion relation of Bogoliubov modes localized near a topological defect - a hard wall - in Bose-Einstein condensate

We consider a Bose-Einstein condensate of bosons with repulsion, described by the Gross-Pitaevskii equation and restricted by an impenetrable "hard wall" (either rigid or flexible) which is intended to suppress the "snake instability" inherent for dark solitons. We solve analytically the Bogoliubov - de Gennes equations to find the spectra of gapless Bogoliubov excitations localized near the "domain wall" and therefore split from the bulk excitation spectrum of the Bose-Einstein condensate. The "domain wall" may model either the surface of liquid helium or of a strongly trapped Bose-Einstein condensate. The dispersion relations for the surface excitations are found for all wavenumbers $k$ along the surface up to the "free-particle" behavior $k \rightarrow \infty$, the latter was shown to be bound to the "hard wall" with some "universal" energy $\Delta$.