We study the analytical properties and the numerical methods for the Bogoliubov-de Gennes equations (BdGEs) describing the elementary excitation of Bose-Einstein condensates around the mean field ground state, which is governed by the Gross-Pitaevskii equation (GPE). Derived analytical properties of BdGEs can serve as benchmark tests for numerical algorithms and three numerical methods are proposed to solve the BdGEs, including sine-spectral method, central finite difference method and compact finite difference method. Extensive numerical tests are provided to validate the algorithms and confirm that the sine-spectral method has spectral accuracy in spatial discretization, while the central finite difference method and the compact finite difference method are second-order and fourth-order accurate, respectively. Finally, sine spectral method is extended to study the elementary excitations under optical lattice potential and solve the BdGEs around the first excited states of the GPE. The numerical experiments demonstrate the efficiency and accuracy of the proposed methods for solving BdGEs.

This is joint work with Yali Gao. This work was supported by NSFC grants U1530401, 11771036, and 91630204.