 Bogoliubov treatment of the Gross–Pitaevskii equation is a method that finds the elementary excitations of a Bose–Einstein condensate.To that purpose, the condensate wavefunction is approximated by a sum of the equilibrium wavefunction formula_42 and a small perturbation formula_43,

Then this form is inserted in the time dependent Gross–Pitaevskii equation and its complex conjugate, and linearized to first order in formula_43

Assuming the following for formula_43

one finds the following coupled differential equations for formula_50 and formula_51 by taking the formula_52 parts as independent components

For a homogeneous system, i.e.for formula_55, one can get formula_56 from the zeroth order equation.