Stability of the solutions of the Gross-Pitaevskii equation

We examine the static and dynamic stability of the solutions of the Gross-Pitaevskii equation and demonstrate the intimate connection between them. All salient features related to dynamic stability are reflected systematically in static properties. We find, for example, the obvious result that static stability always implies dynamic stability and present a simple explanation of the fact that dynamic stability can exist even in the presence of static instability.

Figure 1

Eigenvalues of the static and dynamic stability matrices as functions of the coupling constant $g$, computed by means of exact diagonalization in a truncated basis. Solid lines represent the imaginary part of the eigenvalues $\mathrm{Im}\left({\omega }_d\right)$ of the dynamic stability matrix. Dots represent the lowest eigenvalues ${\overline{\omega }}_s$ of the static stability matrix.