非零温度下三次和五次非线性BEC的NLS方程的第一性原理推导线性波频散
First principles derivation of NLS equation for BEC with cubic and quintic nonlinearities at non zero temperature. Dispersion of linear waves
摘要：
在这项工作中，我们提出了一个中性玻色子的量子流体动力学方程的推导。我们考虑了粒子间的短程相互作用。这种相互作用包括二元相互作用$U(\textbf{r}_{i},\textbf{r}_{j})$和三粒子相互作用$U(\textbf{r}_{i},\textbf{r}_{j},\textbf{r}_{k})$，后者不包括粒子间的二元相互作用。从玻色-爱因斯坦凝聚体的量子流体力学方程出发，导出了非线性薛定谔方程。该方程包括三阶和五阶非线性。它处于零度。采用了三粒子相互作用常数的显式形式。首先，发展了研究线性波频散的方法。比较了线性波的频散特性。在零温下，两粒子间的相互作用近似于相互作用半径的三级（TOIR），三粒子间的相互作用近似于三粒子间的相互作用。我们考虑了温度对元激发色散的影响。为此目的，我们推导出一个系统的QHD方程在非零温度。所得方程组是众所周知的双流体流体力学的模拟。此外，它是由于三粒子相互作用的两流体流体力学方程的推广。计算了第一和第二声速通过超流和非凝结组分浓度的明显表达式。
英文摘要：
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 $U(\textbf{r}_{i},\textbf{r}_{j})$ and three particle interaction $U(\textbf{r}_{i},\textbf{r}_{j},\textbf{r}_{k})$, the last one does not include binary interaction between particles. From the quantum hydrodynamic (QHD) equations for Bose-Einstein condensate we derive nonlinear Schrödinger 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.
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