  We discuss magnetoexcitons dispersion in single and coupled $GaAs-(Ga,Al)As$
quantum wells using the Bethe-Salpeter (B-S) formalism. The B-S formalism in
the case of quantum wells provides an equation for the exciton wave function
which depends on two space variables plus the time variable, i.e. the B-S
equation is 2+1-dimensional equation. We compare the results for
magnetoexcitons dispersion, obtained in the LLL approximations with the results
calculated by solving the exact B-S equation. It is shown that the exact B-S
equation has an extra term (B-S term) that does not exist in the LLL
approximation. Within the framework of the variational method, we obtain that,
(i) the ground-state energy of a heavy-hole magnetoexciton with a zero wave
vector in $GaAs-(Ga,Al)As$ quantum wells, calculated by means of the exact B-S
equation, is very close to the ground-state energy, obtained in the LLL
approximation, (ii) in a strong perpendicular magnetic field the magnetoexciton
dispersion (in-plane magnetoexciton mass) is determined mainly by the B-S term
rather than the term that describes the electron-hole Coulomb interaction in
the LLL approximation.
