I'm not sure what your question is here. For a large class of BEC problems, the GP mean-field theory is known to be an excellent approximation. It is also a relatively simple description that is easy to work with, so from that perspective there is little reason not to use it, unless we are specifically interested in effects beyond the GPE.

But K-G equation is equally powerful with Landau potential in the Hamiltonian. So, why people do not like to accept it and be biased, although GPE is still an useful means, as suggested by Magnus.

The dilute gas condensates currently studied in laboratories are nonrelativistic and electrically neutral, so I think the Klein-Gordon equation would reduce to its nonrelativistic limit, the Schroedinger equation, and the usual minimal-coupling description of the interaction of charges with a magnetic vector potential would be unwieldy to say the least!  The GP equation is actually the beginning, not the end, of modern treatments of degenerate Bose-Eisnstein gases; it describes the pure-condensate, zero-temperature limit, and can be built upon using quantum many-body techniques.  It happens to work remarkably well in cases where the non-condensate fraction of the gas is small, 0.1 or less, as has been demonstrated by much detailed comparison of first-principles theory with experiment.

Thanks a lot to: Charles

For molecular BEC (non-dilute) with multispecies, K-G equ seems to work with nonlinear Landau potential. Via Stat mech, when the effective m,ass is derived, it is found that at 0K, its infinity - so the bosons or solitons condense in the bound state, where only Bright Solitons are present. Without invoking GPE, this is shown categorically which seems to be difficult to be accepted by many people. WHY??

Could you perhaps provide links or references to the work which you describe so that we may read about it?

Cheers, Gordon.