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

In this work we presented a derivation of the quantum hydrodynamic equations for neutral bosons. We considered short range interaction between particles. This interaction consist binary interaction $U({\mathbf{\text{r}}}_i,{\mathbf{\text{r}}}_j)$ and three particle interaction $U({\mathbf{\text{r}}}_i,{\mathbf{\text{r}}}_j,{\mathbf{\text{r}}}_k)$, the last one does not include binary interaction between particles. From the quantum hydrodynamic (QHD) equations for Bose-Einstein condensate we derive nonlinear Schr"{o}dinger equation. This equation includes the nonlinearities of third and fifth degree. It is at zero temperature. Explicit form of the constant of three-particle interaction was taken. First of all, developed method we used for studying of dispersion of linear waves. Dispersion characteristics of linear waves were compared for the cases. It were of two-particle interaction in approximation third order to interaction radius (TOIR) and three-particle interaction, at zero temperature. We consider influence of temperature on dispersion of elementary excitations. For this aim we derive a system of QHD equations at non-zero temperature. Obtained system of equation is an analog of well-known two-fluid hydrodynamics. Moreover, it is generalization of two-fluid hydrodynamics equations due to three-particle interaction. Evident expressions of the velocities of the first and second sound via the concentrations of superfluid and noncondesate components is calculated.