FIRST PRINCIPLES DERIVATION OF NLS EQUATION FOR BEC WITH CUBIC AND QUINTIC NONLINEARITIES AT NONZERO TEMPERATURE: DISPERSION OF LINEAR WAVES

We present a derivation of the quantum hydrodynamic (QHD) equations for neutral bosons. We consider the short-range interaction between particles. This interaction consist of a binary interaction U(r(i), r(j)) and a three-particle interaction (TPI) U(r(i), r(j), r(k)) and the last one does not include binary interaction between particles. From QHD equations for Bose-Einstein condensate we derive a nonlinear Schrodinger equation. This equation was derived for zero temperature and contains the nonlinearities of the third and the fifth degree. Explicit form of the constant of the TPI is obtained. First of all, developed method we used for studying of dispersion of the linear waves. Dispersion characteristics of the linear waves are compared for different particular cases. We make comparison of the two-particle interaction in the third order by the interaction radius (TOIR) and TPI at the zero temperature. We consider influence of the temperature on the dispersion of the elementary excitations. For this aim we derive a system of the QHD equations at nonzero temperature. Obtained system of equation is an analog of the well-known two-fluid hydrodynamics. Moreover, it is generalization of the two-fluid hydrodynamics equations due to TPI. Explicit expressions of the velocities for the first and the second sound via the concentration of superfluid and noncondensate components is calculated.