Singular nonlinearity management for matter-wave solitons in normal and inverted parabolic potentials

We produce a class of solvable Gross-Pitaevskii equations (GPEs), which incorporate the nonlinearity management, a time-dependent factor in front of the cubic term, accounting for the Feshbach resonance in variable magnetic field applied to the Bose-Einstein condensate, and the trapping potential, which may be either static or time-dependent. The GPE is transformed into an equation with a constant nonlinearity coefficient and an additional time-dependent linear term. We present four examples of the nonlinearity-management scenarios which, in proper conjugation with the trapping potential, lead to solvable GPEs. In two cases, the potential is required in the inverted form, which may be a physically meaningful one. In all the cases, the solvable schemes are singular, with the corresponding nonlinearity-enhancement factor diverging at one or multiple moments of time. This singularity may be relevant to the Feshbach resonance. Solvable equations with the normal trapping potential feature multiple singularities (thus limiting the applicability of the GPE to a finite interval of time), while, with the inverted potential, the singularity occurs only at t = 0, validating the equations for 0 < t < ∞. Using the Hirota transform (HT), we construct bright solitons for all solvable cases, and demonstrate that higher-order solitons can be obtained too. Dark solitons are also found, in counterparts of the same models with self-repulsion. In comparison with the previous analysis, a crucial ingredient of the present method is finding the soliton's chirp.