We consider the nonlinear Schrödinger equation describing the dipolar Bose–Einstein condensates with and without harmonic trapping potential. We first show the existence and strong instability of ground state standing waves for the equation in the case of no external potential. We next prove the strong instability of ground state standing waves for the equation in the presence of a general harmonic potential, under a suitable assumption. These results are improvements of recent results whose proofs require an additional condition. Finally we prove the existence and orbital stability of prescribed mass standing waves for the equation with harmonic potential. The proof of the latter result contains some improvements of previous literature.

我们考虑了非线性Schrödinger方程，该方程描述了带有和不带有谐波陷波电势的偶极Bose-Einstein冷凝物。我们首先证明在没有外部电势的情况下方程的基态驻波的存在和强烈不稳定性。接下来，我们在适当的假设下证明了在存在一般谐波电势的情况下，方程的基态驻波的强烈不稳定性。这些结果是对最近结果的改进，其证据需要附加条件。最后，我们证明了具有谐波势的方程所规定的质量驻波的存在性和轨道稳定性。后一个结果的证明包含了先前文献的一些改进。