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Number-conserving approaches to $n$-component Bose-Einstein condensates

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.