Uniform-density Bose-Einstein condensates of the Gross-Pitaevskii equation found by solving the inverse problem for the confining potential

In this work, we study the existence and stability of constant density (flat-top) solutions to the Gross-Pitaevskii equation (GPE) in confining potentials. These are constructed by using the “inverse problem” approach which corresponds to the identification of confining potentials that make flat-top waveforms exact solutions to the GPE. In the one-dimensional case, the exact solution is the sum of stationary kink and antikink solutions, and in the overlapping region, the density is constant. In higher spatial dimensions, the exact solutions are generalizations of this wave function. In the absence of self-interactions, the confining potential is similar to a smoothed-out finite square well with minima also at the edges. When self-interactions are added, terms proportional to $\pm g{\psi }^{*}\psi$ and $\pm gM$ with $M$ representing the mass or number of particles in Bose-Einstein condensates get added to the confining potential and total energy, respectively. In the realm of stability analysis, we find (linearly) stable solutions in the case with repulsive self-interactions which also are stable to self-similar deformations. For attractive interactions, however, the minima at the edges of the potential get deeper and a barrier in the center forms as we increase the norm. This leads to instabilities at a critical value of $M$. Comparing the stability criteria from Derrick's theorem with Bogoliubov-de Gennes (BdG) analysis stability results, we find that both predict stability for repulsive self-interactions and instability at a critical mass $M$ for attractive interactions. However, the numerical analysis gives a much lower critical mass. This is due to the emergence of symmetry-breaking instabilities that were detected by the BdG analysis and violate the symmetry $x\to -x$ assumed by Derrick's theorem.

Figure 1

One-dimensional condensate density $\rho \left(x\right)$ and potentials $V\left(x\right)$ for $q=5$ and $\omega =-4$ for $g=\pm 1$.

Figure 2

The 1D energy per particle as a function of $q$ with $\omega =-4$ and $M=1$ for $g=\pm 1$.

Figure 3

The 1D energy $h\left(\beta \right)$ as a function of $\beta$ for $g=\pm 1$ and for $M_0=1$.

Figure 4

Plot of $h_3(a,M)$ for the 1D case for $g=\pm 1$.

Figure 5

Plot of $h_3\left(a\right)$ for $M=2$ for the 1D case for $g=-1$. The minimum is at $a=0.023$.

Figure 6

$V(x,y)$ for the square flat-top soliton with $q=5$.

Figure 7

The density $\rho (x,y)$ for the square, flat-top soliton with $q=5$ and $M=1$.

Figure 8

Plot of $e\left(q\right)=E\left(q\right)/M$ for the 2D square case with $\omega =-8$, $M=20$, and $g=\pm 1$. The red curve is for the attractive case ($g=-1$).

Figure 9

Plot of $h(\beta ,M)$ for the 2D square case with $g=\pm 1$, ${\omega }_0=-8$, and $M=1,10,20,50$.

Figure 10

$V(x,y)$ given by Eq. (35) for $q=5$.

Figure 11

The radial density $\rho \left(r\right)$ and potentials $V\left(r\right)$ for $g=\pm 1$ for the 2D radial case with $M=20$ and $q=5$.

Figure 12

Plot of $e\left(q\right)$ for the 2D radial case with $\omega =-4$, $q=5$, and $M=20$. The repulsive case ($g=+1$) is given by the blue curve, the attractive case ($g=-1$) is given by the red curve.

Figure 13

The energy of the stretched round 2D flat-top soliton $h\left(\beta \right)$ as a function of $\beta$.

Figure 14

Spectral stability analysis of 1D flat-top soliton solutions for (a) $g=-1$ (attractive) and (b) $g=1$ (repulsive), respectively. The left and right columns depict respectively the imaginary and real parts of the eigenvalues of the stability problem of Eq. (40). The parameter values here are $\omega =-4$ and $q=5$. Note that the parameter $M$ herein coincides with the mass (or $l_2$ norm) of the flat-top soliton solution via Eq. (11).

Figure 15

Top panels: Spectral stability analysis and existence results of bifurcating branches emanating from the flat-top soliton solution in 1D with $g=-1$ (and $q=5$ as well as $\omega =-4$). The left and middle panels depict ${\lambda }_i$ and ${\lambda }_r$ as functions of $M$ (the same spectral picture is obtained for the other branch that has the same norm). Note that the bifurcating branch is spectrally stable due to the absence of real eigenvalues (see the middle panel). The right panel depicts spatial profiles of the density of the bifurcating branches for $M=30$. Note that the density of the flat-top soliton solution for $M=30$ is plotted, too, in the panel with dashed-dotted black lines for comparison. Bottom panels: Spatiotemporal evolution of densities $\rho \left(x\right)$ for the bifurcating branches is shown in the left and middle panels with $M=30$, as well as the flat-top soliton solution (for the same $M$) in the right panel. For the stable steady states, we perturbed the initial condition with a random perturbation [of ${10}^{-3}\times \max \left(\right|u^{\left(0\right)}\left|\right)$ amplitude], whereas for the unstable flat-top soliton solution of the right panel, we perturbed the initial condition by considering the eigenvector corresponding to the most unstable eigendirection.

Figure 16

Spatiotemporal evolution of the density $\rho \left(x\right)$ for a perturbed flat-top soliton solution for (a) $M=0.65$, (b) $M=2$, and (c) $M=4$, with $g=-1$, $q=5$, and $\omega =-4$. For the stable steady state of panel (a), a random perturbation with amplitude ${10}^{-3}\times \max \left(\right|u^{\left(0\right)}\left|\right)$ was added to the localized pulse, whereas for the unstable states of panels (b)–(c), the initial condition was perturbed by the most unstable eigendirection (and with the same amplitude for the pertinent cases).

Figure 17

Spectral stability analysis of 2D square flat-top soliton solutions for (a) $g=-1$ (attractive) and (b) $g=1$ (repulsive). The format of the figure is the same as of Fig. 14. The parameter values here are $\omega =-8$ and $q=5$.

Figure 18

Same as Fig. 17 but for the 2D radial flat-top soliton solutions with (a) $g=-1$ (attractive) and (b) $g=1$ (repulsive). The format of the figure is the same as of Fig. 14. The parameter values here are $\omega =-4$ and $q=5$.

Figure 19

Spatial distribution of the density $\rho (x,y)$ at $t=500$ corresponding to perturbed square (top panels) and radial (bottom panels) flat-top soliton solutions for $g=-1$ (left column) and $g=1$ (right column). The densities shown in the left and right columns correspond to $M=5$ and $M=20$, respectively, i.e., at values of $M$ where the solutions are linearly stable (see Figs. 17 and 18). For the square flat-top solitons, $\omega =-8$, whereas $\omega =-4$ for the radial ones (with $q=5$ in both cases).

Figure 20

Snapshots of densities $\rho (x,y)$ of linearly unstable square (top panels) and radial (bottom panels) flat-top soliton solutions with $M=12$, and $g=-1$. The rest of the parameter values are the same as in Fig. 19.