## 2 The Model  
At ultra-low temperatures, a DBEC with two-body, three-body and nonlocal DD interactions can be described by the following time-dependent GP Equation [1; 2; 23; 24; 25; 58]:  
\[ \begin{split}\mathrm{i}\hbar\frac{\partial\phi(\mathbf{r},t)}{ \partial t}=&\Big{[}-\frac{\hbar^{2}}{2m}\
abla^{2}+V(\mathbf{r })+\frac{4\pi\hbar^{2}a(t)N}{m}|\phi(\mathbf{r},t)|^{2}+N\int U_{\mathrm{dd} }(\mathbf{r}-\mathbf{r}^{\prime})\big{|}\phi(\mathbf{r}^{\prime},t)\big{|}^{2 }d\mathbf{r}^{\prime}\\ &+\eta(t)N^{2}|\phi(\mathbf{r},t)|^{4}\Big{]}\phi(\mathbf{r},t), \end{split}  \tag{1}\]  
where \(\phi(\mathbf{r},t)\) refers to the condensate wave function, \(N\) the number of particles with magnetic dipole moment \(\mu\), \(m\) the mass, \(\
abla^{2}\) the Laplacian operator, \(\hbar\) the reduced Planck constant, \(V(\mathbf{r})=\frac{1}{2}m\Big{(}\omega_{x}^{2}x^{2}+\omega_{y}^{2}y^{2}+ \omega_{z}^{2}z^{2}\Big{)}\) is the trapping potential with frequencies \(\omega_{x},\omega_{y}\) and \(\omega_{z}\), and \(a(t)\) is the time-modulated atomic scattering length. The potential of the dipolar interaction for magnetic dipoles is \(U_{\mathrm{dd}}(\mathbf{R})=\frac{\mu_{0}\hbar^{2}}{4\pi}\frac{1-3\cos^{2} \theta}{|\mathbf{R}|^{3}}\), where \(\mathbf{R}=\mathbf{r}-\mathbf{r}^{\prime}\) determines the relative position of dipoles and \(\theta\) is the angle between \(\mathbf{R}\) and the vertical (\(z\)) polarization direction, \(\mu_{0}\) is the free-space permeability [23]. The parameter \(\eta(t)\) is the strength of the three-body interaction which depends on both the \(s\)-wave scattering length \(a(t)\) and the universal constants \(d_{1},d_{2},a_{0}\) and \(s_{0}\)[58]. This parameter is \(\eta(t)=\frac{12\pi\hbar^{2}a(t)^{2}}{m}\Big{(}d_{1}+d_{2}\,tan\Big{[}s_{0}\, ln\frac{|a(t)|}{|a_{0}|}+\frac{\pi}{2}\Big{]}\Big{)}\) or \(\eta(t)=\frac{12\pi\hbar^{2}a(t)^{2}}{m}\Big{(}d_{1}-d_{2}\,cot\Big{[}s_{0}\, ln\frac{|a(t)|}{|a_{0}|}\Big{]}\Big{)}\), where the numerical values of the universal constants are given in Refs. [58; 59]. However, the strength of the three-body interaction is very small when compared with the strength of the two-body interaction [60]. In some other regimes, the three-body interaction may become a dominant factor under special conditions as in the cases of higher densities and Effimov resonance [58]. The normalization is  
\[\int d\mathbf{r}|\phi(\mathbf{r},t)|^{2}=1. \tag{2}\]  
To compare the dipolar and contact interactions, it is often useful to introduce the length scale \(a_{\rm dd}\equiv\mu_{0}\mu^{2}m/(12\pi\hbar^{2})\), whose experimental values for \({}^{52}\)Cr, \({}^{164}\)Er and \({}^{168}\)Dy condensates are \(16a_{0}\), \(66a_{0}\) and \(131a_{0}\), respectively, \(a_{0}\) being the Bohr radius [1,2]. In the present investigation, we use the parameters of \({}^{52}\)Cr. It is convenient to rescale Equation (1) into a dimensionless form. For this purpose, we apply the transformation \(\tilde{\mathbf{r}}=\mathbf{r}/l,\bar{\mathbf{R}}=\mathbf{R}/l,\bar{a}(t)=a(t)/ l,\bar{a}_{\rm dd}=a_{\rm dd}/l,\tilde{t}=t\tilde{\omega}\), \(\tilde{x}=x/l,\tilde{y}=y/l,\tilde{z}=z/l,\tilde{\varphi}=l^{3/2}\phi\), where the harmonic-oscillator length is \(l=\sqrt{\hbar/(m\tilde{\omega})}\)and rewrite Equation (1) (dropping the overbar) as  
\[ \begin{split} i\frac{\partial\phi(\mathbf{r},t)}{\partial t}=& \bigg{[}-\frac{1}{2}\
abla^{2}+V(\mathbf{r})+4\pi a(t)N|\phi( \mathbf{r},t)|^{2}+3 Na_{\rm dd}\int\frac{1-3\cos^{2}\theta}{|\mathbf{R}|^{3}}| \phi(\mathbf{r}^{\prime},t)|^{2}d\mathbf{r}^{\prime}\\ &+\chi(t)|\phi(\mathbf{r},t)|^{4}\bigg{]}\phi(\mathbf{r},t),\end{split}  \tag{3}\]  
where \(\chi(t)=\frac{\eta(t)N^{2}}{m\tilde{\omega}^{2}\hbar}\),  
\[V(\mathbf{r})=\frac{1}{2}(\gamma^{2}x^{2}+\
u^{2}y^{2}+\lambda^{2}z^{2}), \tag{4}\]  
and \(\gamma=\omega_{x}/\tilde{\omega},\
u=\omega_{y}/\tilde{\omega},\lambda= \omega_{z}/\tilde{\omega}\), where \(\tilde{\omega}=(\omega_{x}\omega_{y}\omega_{z})^{1/3}\).