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correct result, the inclusion of only one more Gaussian brings the chemical potential curve already close to the numerical result, and practically no improvement is vis- ible in Fig. 1 when 3 or more Gaussians are included. Using 5 coupled Gaussians the exact bifurcation point is reproduced with an accuracy of 10 − $^{6}$. Similar results are obtained in the presence of a trapping potential. 

We now turn to dipolar condensates. Previous studies [12, 16] have shown that in certain regions of the param- eter space dipolar condensates assume a non-Gaussian biconcave “blood-cell-like” shape. To demonstrate the power of the coupled Gaussian wave packet method, we choose a set of such parameters. We consider an axisym- metric trap with (particle number scaled) trap frequen- cies N $^{2}$γ$_{z}$ = 25200 along the polarization direction of the dipoles and N $^{2}$γ$_{ρ}$ = 3600 in the plane perpendicular to it (corresponding to an aspect ratio of λ = γ$_{z}$/γ$_{ρ}$ = 7). For this set of parameters we show in Fig. 2 (a) the 

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convergence behavior of the mean field energy. We com- pare the variational solution as the number of Gaussian wave packets is increased from 2 to 6 with the mean field energy value of a numerical lattice calculation (imagi- nary time evolution combined with FFT) with a grid size of 128 × 512, at scattering length a = 0 as an example. The mean field energy for one Gaussian is E$_{mf}$ = 60361 E$_{d}$ and lies far outside the vertical energy scale. Evidently the numerical value is more than excel- lently reproduced by 5 and 6 coupled Gaussians. The behavior for other scattering lengths is similar. Also the wave function nicely converges, and moreover, as can be 

seen in Fig. 2 (b), reproduces the biconcave shape of the condensate as does the numerical solution. Thus the method of coupled Gaussians is a viable and full-fledged alternative to direct numerical solutions of the Gross- Pitaevskii equation for dipolar condensates. 

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Figure 3 (a) shows, for the same set of trap frequen- cies, the results for the mean field energy of the conden- sate as a function of the scattering length a (in units a$_{d}$ ) for a wave function with one Gaussian, and for 5 cou- pled Gaussian wave packets. Results obtained using 6 Gaussians would be indistinguishable in the figure from those obtained using 5 Gaussians, and the results for 2– 4 Gaussians are not shown for the sake of clarity of the figure. 

Similar to the above findings for monopolar conden- sates, and as is known from previous variational calcula- tions [9] for dipolar condensates, for N = 1 two branches of solution are born in a tangent bifurcation. The en- 

