Firstly I am not sure as to whether I am using the word "condensate" in the right context. In QFT contexts I think I see it getting used to mean the space-time independent solution which would solve the Euler-Lagrange equations of the action that would sit in the exponent in the path-integral - which in general might be different from the classical action. I would like to know why are these kinds of solutions so important - because this is picking out some special configurations among the entire space of classical solutions which would in general include non-trivially dynamical solutions.
Now when one is doing a "small" fluctuation about the condensate and integrating out degrees of freedom to get an effective action for one of the fluctuation variables then there are two issues which confuse me -
In multi component fields (like say complex ones which can be thought of as the modulus and the phase) what drives the choice as to which fluctuation is to be integrated out? (..in the complex case I guess in general people talk to the effective action for the phase fluctuation..)
What is most confusing to me is to understand how to determine whether the space-time derivatives of the fluctuations are big or small. If one is doing the calculation to say second order then does one keep the products and squares of the derivatives of the fluctuation at the same level of perturbation as the squares and products of the fluctuation themselves? I can't see a natural scale for the derivatives of the fluctuations to which I can compare the derivatives to decide whether they are large or small.
The condensate state is not just some solution of the classical equations of motion. It is the one which has the lowest energy. Since bosons like to be in the same quantum state, at low temperature they macroscopically occupy this lowest state. This is know as formation of the condensate.
The condensate wave-function is rigid in the sense that one needs some macroscopic amount of energy for even a small change of its modulus. However, it cost very little to disturb its phase. Thus at low energies it is natural to construct the effective theory of the phase variations (Goldstone bosons). Usually one integrates out the fluctuations of the gapped modes which leads to a local action of Goldstones only. In a conventional superconductor, for example, one would integrate out the fermions (which are gapped around the Fermi surface) and the fluctuations of the modulus of the condensate.
I think there is no unique way how to assign the smallness order to different spatial and temporal derivatives of the Goldtones. In other words, there is an ambiguity in the choice of power counting for the given effective theory. The power-counting scheme determines which energy and momentum regime the effective theory is applicable to. As far as I understand, however, Gaussian formula allows to integrate out any action quadratic in the gapped fluctuations, so this can be done exactly.