  We study nonlinear mixing effects among quadrupole modes and scissors modes
in a harmonically trapped Bose-Einstein condensate. Using a perturbative
technique in conjunction with a variational approach with a Gaussian trial wave
function for the Gross-Pitaevskii equation, we find that mode mixing
selectively occurs. Our perturbative approach is useful in gaining qualitative
understanding of the recent experiment [Yamazaki et al., J. Phys. Soc. Japan
84, 44001 (2015)], exhibiting a beating phenomenon of the scissors mode as well
as a modulation phenomenon of the low-lying quadrupole mode by the high-lying
quadrupole mode frequency. Within the second-order treatment of the nonlinear
mode coupling terms, our approach predicts all the spectral peaks obtained by
the numerical simulation of the Gross-Pitaevskii equation.
