Large amplitude spatial fluctuations in the boundary region of the Bose–Einstein condensate in the Gross–Pitaevskii régime

The Gross–Pitaevskii régime of a Bose–Einstein condensate is investigated using a fully non-linear approach. The confining potential first adopted is that of a linear ramp. An infinite class of new analytical solutions of this linear ramp potential approximation to the Gross–Pitaevskii equation is found which are characterised by pronounced large-amplitude oscillations close to the boundary of the condensate. The limiting case within this class is a nodeless ground state which is known from recent investigations as an extension of the Thomas–Fermi approximation. We have found the energies of the oscillatory states to lie above the ground state energy but recent experimental work, especially on spatially confined superconductors, indicates that such states may be easily occupied and made manifest at finite temperatures. We have also investigated their stability using a Poincaré section analysis as well as a linear perturbation approach. Both these techniques demonstrate stability against small perturbations. Finally, we have discussed the relevance of these quasi-one-dimensional solutions in the context of the fully three-dimensional condensates. This has been argued on the basis of numerical work and asymptotic approximations.