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The nonlinear dynamics of self-bound Bose-Einstein matter waves under the action of an attractive nonlocal and a repulsive local interaction is analyzed by means of a time-dependent variational formalism. The mean-field model described by the Gross-Pitaevskii equation is reduced to a single second-order conservative ordinary differential equation admitting a potential function. The stable fixed points and the linear oscillation frequencies of the breather modes are determined. The chosen nonlocal interactions are a Gaussian-shaped potential and a van der Waals-like potential. The variational solutions are compared with direct numerical simulations of the Gross-Pitaevskii equation, for each of these nonlocalities. The spatio-temporal frequency spectra of linear waves are also determined.

Time-dependent variational approach for Bose–Einstein condensates with nonlocal interaction