  We analyze possible nonlinear exciton-exciton correlation effects in the
optical response of semiconductors by using a time-dependent density-functional
theory (TDDFT) approach. For this purpose, we derive the nonlinear
(third-order) TDDFT equation for the excitonic polarization. In this equation,
the nonlinear time-dependent effects are described by the time-dependent
(non-adiabatic) part of the effective exciton-exciton interaction, which
depends on the exchange-correlation (XC) kernel. We apply the approach to study
the nonlinear optical response of a GaAs quantum well. In particular, we
calculate the 2D Fourier spectra of the system and compare it with experimental
data. We find that it is necessary to use a non-adiabatic XC kernel to describe
excitonic bound states - biexcitons, which are formed due to the retarded TDDFT
exciton-exciton interaction.
