  We study theoretically the ultrafast nonlinear optical response of quantum
well excitons in a perpendicular magnetic field. We show that for
magnetoexcitons confined to the lowest Landau levels, the third-order
four-wave-mixing (FWM) polarization is dominated by the exciton-exciton
interaction effects. For repulsive interactions, we identify two regimes in the
time-evolution of the optical polarization characterized by exponential and
{\em power law} decay of the FWM signal. We describe these regimes by deriving
an analytical solution for the memory kernel of the two-exciton wave-function
in strong magnetic field. For strong exciton-exciton interactions, the decay of
the FWM signal is governed by an antibound resonance with an
interaction-dependent decay rate. For weak interactions, the continuum of
exciton-exciton scattering states leads to a long tail of the time-integrated
FWM signal for negative time delays, which is described by the product of a
power law and a logarithmic factor. By combining this analytic solution with
numerical calculations, we study the crossover between the exponential and
non-exponential regimes as a function of magnetic field. For attractive
exciton-exciton interaction, we show that the time-evolution of the FWM signal
is dominated by the biexcitonic effects.
