In this paper, we analyze and discuss instabilities and self-oscillations of four-wave mixing in two-component Bose–Einstein condensates with two- and three-body interatomic interactions. The model is very accurately described in the mean-field approximation by the cubic–quintic Gross–Pitaevskii equation. The relation between the input and output field intensities is multivalued and the effects of the quintic nonlinearity on the self-oscillations of the system are studied. We have also found that the magnitude of the signal beam increases with the increase of the intensity of the probe beam, up to a saturated value, then it decreases with the increase of the intensity of the probe beam. We have shown that the three-body interatomic interactions enhance this saturated value.

Modulation instabilities of matter waves described by a system of four coupled Gross–Pitaevskii equations with two-and three-body interactions are analytically and numerically investigated. For analytical treatment, we use the linear stability analysis to derive the dispersion relation which allows us to predict regions of modulation instabilities gain spectra in two regimes. In the first regime, two counter-propagating beams of matter waves intensities are different, while in the second, their intensities are equal. It is shown that both the two-and three-body interactions greatly affect the gain spectra in the two regimes. By full numerical simulations of four coupled Gross–Pitaevskii equations, we confirm the analytical predictions. Moreover, we explore with the same parameters two possibilities of modulation instabilities for different velocities v that depend on the two-body interactions. For the small values of two-body interactions, when v > 1, the spatiotemporal evolution appears to be stable for long times, the modulation instability is enhanced. On the contrary, when v < 1, this evolution becomes unstable, the modulation instability tends to be suppressed.

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Recent research works on ultra cold quantum gases demonstrated that dipolar Bose–Einstein condensates (BECs) exhibit rich spatiotemporal dynamic where both local and nonlocal interactions are considered. We explore theoretically the possibility of controlling the formation and dynamics of soliton molecules in binary dipolar condensates with spin-orbit coupling (SOC). We exploit the variational technique to derive the new equations of motion for the widths and amplitudes, the effective potential and the oscillation frequency of the molecules. Our study confirms the existence of stable localized bound states in an optical potential. We find that the integrity of the molecules is influenced by the physical parameters, notably the local and nonlocal interactions with the SOC. These parameters are carefully chosen by the Vakhitov–Kolokolov (VK) criterion to ensure the stability of the molecules. We present the results of numerical experiments and confirm the analytical predictions. Moreover, we show the soliton–soliton interaction in each molecule when the local interactions are strong.