## 3 The Variational Method
When \(\Omega\gg 1\), \(R_{1}(t)\) is small of the order of \(\sim\)\(\Omega^{-2}\)Keeping the terms of the order of \(\Omega^{-2}\) in \(R_{1}(t)\), one may obtain the following equations of motion for \(R_{0}(t)\) and \(R_{1}(t)\),  
\[\frac{d^{2}R_{1}(t)}{dt^{2}}= \frac{\sqrt{2}N\epsilon_{1}\sin\left(\Omega t\right)}{\sqrt{\pi}d_ {z}R_{0}^{3}}+\frac{4N^{2}\chi_{1}\sin\left(\Omega t\right)}{9\sqrt{3}d_{z}^{2 }\pi^{3}R_{0}^{5}} \tag{15}\] \[\frac{d^{2}R_{0}(t)}{dt^{2}}= \frac{1}{R_{0}^{3}}+\frac{N[2\epsilon_{0}-a_{dd}\,\Lambda(\xi_{0} (t))]}{\sqrt{2\pi}d_{z}\,R_{0}^{3}}+\frac{4N^{2}\chi_{0}}{9\sqrt{3}\pi^{3}d_{z }^{2}\,R_{0}^{5}}-\frac{3\sqrt{2}N\epsilon_{1}<R_{1}\sin(\Omega t)>}{\sqrt{\pi }d_{z}\,R_{0}^{4}}\] (16) \[\mathcal{L} = \frac{i}{2}\bigg{(}\psi\frac{\partial\psi^{*}}{\partial t}-\psi^{ *}\frac{\partial\psi}{\partial t}\bigg{)}+\frac{|\
abla_{\rho}\psi|^{2}}{2}+ \frac{2\pi a(t)N}{\sqrt{2\pi}d_{z}}|\psi|^{4}+\frac{\chi(t)}{3\sqrt{3}\pi d_{ z}^{2}}|\psi|^{6} \tag{9}\]0  
where the \(<R_{1}\sin(\Omega t)>\) indicates the time average of the rapid oscillation. From Equation (15), we obtain  
\[\mathcal{L} = \frac{i}{2}\bigg{(}\psi\frac{\partial\psi^{*}}{\partial t}-\psi^{ *}\frac{\partial\psi}{\partial t}\bigg{)}+\frac{|\
abla_{\rho}\psi|^{2}}{2}+ \frac{2\pi a(t)N}{\sqrt{2\pi}d_{z}}|\psi|^{4}+\frac{\chi(t)}{3\sqrt{3}\pi d_{ z}^{2}}|\psi|^{6} \tag{9}\]1  
We now substitute \(R_{1}(t)\) into Equation (16) to obtain the following equation of motion for \(R_{0}(t)\)  
\[\mathcal{L} = \frac{i}{2}\bigg{(}\psi\frac{\partial\psi^{*}}{\partial t}-\psi^{ *}\frac{\partial\psi}{\partial t}\bigg{)}+\frac{|\
abla_{\rho}\psi|^{2}}{2}+ \frac{2\pi a(t)N}{\sqrt{2\pi}d_{z}}|\psi|^{4}+\frac{\chi(t)}{3\sqrt{3}\pi d_{ z}^{2}}|\psi|^{6} \tag{9}\]2 \[\mathcal{L} = \frac{i}{2}\bigg{(}\psi\frac{\partial\psi^{*}}{\partial t}-\psi^{ *}\frac{\partial\psi}{\partial t}\bigg{)}+\frac{|\
abla_{\rho}\psi|^{2}}{2}+ \frac{2\pi a(t)N}{\sqrt{2\pi}d_{z}}|\psi|^{4}+\frac{\chi(t)}{3\sqrt{3}\pi d_{ z}^{2}}|\psi|^{6} \tag{9}\]3  
The VA suggests that the effect of the DD interaction is to minimize the impact of the constant part of the contact interaction for \(a_{dd}>0\). Thus, one can conclude that the system may become effectively attractive depending on the choice of \(a_{dd}\) and \(\epsilon_{0}\). Hence one can explore the possibility of creation of bright solitons even for positive (repulsive) scattering lengths provided \(a_{dd}>\epsilon_{0}\). Thus, it is obvious that DD interaction minimizes the effect of contact interaction as dipolar BECs really need non-zero dipolar strength to compete with contact interaction. In addition, DD interaction (\(a_{d}d>0\)) always counteracts with two-body contact interaction. We do agree under the present circumstances that the dipoles assume a pancake that are oriented in the z-direction and dipolar BEC expands along the xy plane. To arrest the expansion, here we use the attractive two-body contact interaction. The effective potential \(U(R_{0})\) corresponding to the above equation of motion can be written as  
\[\mathcal{L} = \frac{i}{2}\bigg{(}\psi\frac{\partial\psi^{*}}{\partial t}-\psi^{ *}\frac{\partial\psi}{\partial t}\bigg{)}+\frac{|\
abla_{\rho}\psi|^{2}}{2}+ \frac{2\pi a(t)N}{\sqrt{2\pi}d_{z}}|\psi|^{4}+\frac{\chi(t)}{3\sqrt{3}\pi d_{ z}^{2}}|\psi|^{6} \tag{9}\]4 \[\mathcal{L} = \frac{i}{2}\bigg{(}\psi\frac{\partial\psi^{*}}{\partial t}-\psi^{ *}\frac{\partial\psi}{\partial t}\bigg{)}+\frac{|\
abla_{\rho}\psi|^{2}}{2}+ \frac{2\pi a(t)N}{\sqrt{2\pi}d_{z}}|\psi|^{4}+\frac{\chi(t)}{3\sqrt{3}\pi d_{ z}^{2}}|\psi|^{6} \tag{9}\]5  
where \(\delta(\xi_{0})=[1+2\chi_{0}^{2}-3\zeta_{0}^{2}d(\xi_{0})]/\left(1-\xi_{0}^{2}\right)\) and \(\zeta_{0}=R_{0}/d_{z}\). Now, one can scrutinize the nature of the effective potential for different system parameters, namely the strengths of the two-body, three-body and DD interactions. In the present analysis, we explore the possibility of stabilizing the trapless dipolar condensates by manipulating the constant and oscillatory part of the two- and three-body interactions.