Pattern formations in miscellaneous mixtures of Bose-Einstein condensates and the higher-dimensional time-gated Manakov system

In this article, we investigate the structure and dynamics of miscellaneous mixtures of Bose-Einstein condensates confined within a time-independent anisotropic parabolic trap potential. In the zero-temperature mean-field approximation leading to coupled Gross-Pitaevskii equations for the macroscopic wave functions of the condensates, we show that these equations can be mapped onto the higher-dimensional time-gated Manakov system up to a first-order of accuracy. Paying particular attention to two-species mixtures and looking forward deriving a panel of miscellaneous excitations to the above equations, we analyze the singularity structure of the system by means of Weiss et al.'s [J. Weiss, M. Tabor, and G. Carnevale, J. Math. Phys. 24, 522 (1983); 25, 13 (1984).] methodology and provide its general Lax representation. As a result, we unearth a typical spectrum of localized and periodic coherent patterns while depicting elastic and nonelastic interactions among such structures alongside the splitting and resonance phenomena occurring during their motion.