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In this work we presented a derivation of the quantum hydrodynamic equationsfor neutral bosons. We considered short range interaction between particles.This interaction consist binary interaction $U(\textbf{r}_{i},\textbf{r}_{j})$and three particle interaction$U(\textbf{r}_{i},\textbf{r}_{j},\textbf{r}_{k})$, the last one does notinclude binary interaction between particles. From the quantum hydrodynamic(QHD) equations for Bose-Einstein condensate we derive nonlinearSchr\"{o}dinger equation. This equation includes the nonlinearities of thirdand fifth degree. It is at zero temperature. Explicit form of the constant ofthree-particle interaction was taken. First of all, developed method we usedfor studying of dispersion of linear waves. Dispersion characteristics oflinear waves were compared for the cases. It were of two-particle interactionin approximation third order to interaction radius (TOIR) and three-particleinteraction, at zero temperature. We consider influence of temperature ondispersion of elementary excitations. For this aim we derive a system of QHDequations at non-zero temperature. Obtained system of equation is an analog ofwell-known two-fluid hydrodynamics. Moreover, it is generalization of two-fluidhydrodynamics equations due to three-particle interaction. Evident expressionsof the velocities of the first and second sound via the concentrations ofsuperfluid and noncondesate components is calculated.

First principles derivation of NLS equation for BEC with cubic and quintic nonlinearities at non zero temperature. Dispersion of linear waves