Nambu-Goldstone modes in two segregated bose-einstein condensates limited by two hard walls

The Nambu-Goldstone (NG) modes in the system of two segregated Bose-Einstein condensates (BECs) limited by two hard walls are studied by means of the Gross-Pitaevskii (GP) theory. Based on the double-parabola approximation (DPA) combining with the Bogoliubov-de Gennes (BdG) equations we found four NG modes that proves the failure of the Watanabe-Brauner counting rule and, furthermore, their dispersion relations depend explicitly on the geometrical structure.