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Numerical Algorithms

Continuation and stability analysis for Bloch waves of the Gross-Pitaevskii equation

H.-S. Chen, S.-L. Chang, B.‐W. Jeng, C.‐S. Chien

Published May 26, 2017• 3 citations
Publisher
DOI
10.1007/s11075-017-0336-7

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We investigate the band structure of a Bose-Einstein condensate in a one-dimensional periodic potential by calculating stationary solutions of the Gross-Pitaevskii equation, which have the form of Bloch waves. We demonstrate that loops (``swallow tails'') in the band structure occur both at the Brillouin zone boundary and at the center of the zone, and they are therefore a generic feature. A physical interpretation of the swallow tails in terms of periodic solitons is given. The linear stability of the solutions is investigated as a function of the strength of the mean-field interaction, the magnitude of the periodic potential, and the wave vector of the condensate. The regions of energetic and dynamical stability are identified by considering the behavior of the Gross-Pitaevskii energy functional for small deviations of the condensate wave function from a stationary state. It is also shown how for long-wavelength disturbances the stability criteria may be obtained within a hydrodynamic approach.
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. 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 quantity of quantum droplets generated during the nonlinear phase 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 linear stability analysis of the modified Gross-Pitaevskii equation, considering quantum fluctuations, precisely forecasts modulational instability phenomena across different domains of parameter spaces.