Expansion Dynamics of a Two-Component Quasi-One-Dimensional Bose-Einstein Condensate: Phase Diagram, Self-Similar Solutions, and Dispersive Shock Waves

Abstract: We investigate the expansion dynamics of a Bose-Einstein condensate that consists of two components and is initially confined in a quasi-one-dimensional trap. We classify the possible initial states of the two-component condensate by taking into account the non-uniformity of the distributions of its components and construct the corresponding phase diagram in the plane of nonlinear interaction constants. The differential equations that describe the condensate evolution are derived by assuming that the condensate density and velocity depend on the spatial coordinate quadratically and linearly, respectively, what reproduces the initial equilibrium distribution of the condensate in the trap in the Thomas-Fermi approximation. We obtained self-similar solutions of these differential equations for several important special cases and wrote out asymptotic formulas describing the condensate motion on long time scales, when the condensate density becomes so low that the interaction between atoms can be neglected. The problem on the dynamics of immiscible components with the formation of dispersive shock waves was also considered. We compare the numerical solutions of the Gross-Pitaevskii equations with their approximate analytical solutions and study numerically the situations when the analytical method admits no exact solutions.