Wave kinetic description of Bogoliubov oscillations in the Bose-Einstein condensate

A wave kinetic equation, exactly equivalent to the Gross-Pitaevskii equation, describing the dynamics of bosons is presented. The resulting wave kinetic description is used to study the Bogoliubov oscillations of the Bose-Einstein condensate (BEC). An exact quantum dispersion relation and an expression for Landau damping of sound waves in the BEC are derived. Conditions for wave instability are also established. The range of validity of previous quasi-classical results is clarified.