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We develop the number-conserving approach that has previously been used in a single component Bose-Einstein condensed dilute atomic gas, to describe consistent coupled condensate and noncondensate number dynamics, to an $n$-component condensate. The resulting system of equations comprises, for each component, of a generalised Gross-Pitaevskii equation coupled to modified Bogoliubov-de Gennes equations. Lower-order approximations yield general formulations for multi-component Gross-Pitaevskii equations, and systems of multi-component Gross-Pitaevskii equations coupled to multi-component modified number-conserving Bogoliubov-de Gennes equations. The analysis is left general, such that, in the $n$-component condensate, there may or may not be mutually coherent components. An expansion in powers of the ratio of noncondensate to condensate particle numbers for each coherent set is used to derive the self-consistent, second-order, dynamical equations of motion. The advantage of the analysis developed in this article is in its applications to dynamical instabilities that appear when two (or more) components are in conflict and where a significant noncondensed fraction of atoms is expected to appear.
While the Gross--Pitaevskii equation is well-established as the canonical dynamical description of atomic Bose-Einstein condensates (BECs) at zero-temperature, describing the dynamics of BECs at finite temperatures remains a difficult theoretical pro
We present a self-consistent study of coherently coupled two-component Bose-Einstein condensates. Finite spin-flipping coupling changes the first order demixing phase transition for Bose-Bose mixtures to a second order phase transition between an unp
We experimentally investigate the dynamics of spin solitary waves (magnetic solitons) in a harmonically trapped, binary superfluid mixture. We measure the in-situ density of each pseudospin component and their relative local phase via an interferomet
The miscibility of two interacting quantum systems is an important testing ground for the understanding of complex quantum systems. Two-component Bose-Einstein condensates enable the investigation of this scenario in a particularly well controlled se
We systematically construct a series of vector solitary waves in harmonically trapped one-dimensional three-, four-, and five-component Bose-Einstein condensates. These stationary states are continued in chemical potentials from the analytically trac