Modulational instability in quasi-one-dimensional Bose-Einstein condensates

In this work, we investigate the modulational instability of plane wave solutions within a modified Gross-Pitaevskii equation framework. The equation features cubic and quartic nonlinearity. It models the behaviour of quasi-one-dimensional Bose-Einstein condensates in symmetric Bose-Bose mixtures of ultra-dilute cold atoms. The cases of equal amplitudes in both species, as well as of unequal amplitudes are studied. Our study demonstrates the pivotal role of the competition between mean-field attractions and quantum fluctuation-induced repulsions. This competition significantly affects the emergence and evolution of modulational instability. By employing linear stability analysis, we identify the essential conditions that lead to modulational instability. We find that the stability of plane wave solutions significantly depends on the interaction among system parameters. Further development of the instability leads to the fragmentation of the BEC into a chain of quantum droplets. We calculated the number of quantum droplets generated during the nonlinear stage of the instability. Our analytical results are corroborated by numerical simulations of the modified quasi-1D Gross-Pitaevskii equation. These simulations vividly depict the formation, interaction, and coalescence of droplets during the nonlinear phase of modulational instability. The investigation shows that the linear stability analysis of the modified Gross-Pitaevskii equation, considering quantum fluctuations, precisely predicts the initial stage of modulational instability phenomena across different domains of parameter space. In contrast, the later nonlinear stage, studied through numerical simulations, reveals the formation and nonintegrable dynamics of quantum droplets.