In recent years,much attention has been paid to the study of nonlinear excitations in trapping Bose gases due to the remarkable experimental realization of Bose-Einstein condensation (BEC) of cold atomic gases.The most spectacular experimental progress for achieved recently are the observation of solitons and vortices.On the other hand,there is growing interest on the mode-mode resonant interactions of the excitations in trapped condensed Bose gases.In this talk,we shall report our recent research on three-mode interaction and second-harmonic generation in BECs.Based on Gross-Pitaevskii equation we derive a set of nonlinearly coupled envelope equations for a three-mode resonant interaction by means of a method of multiple-scales.We calculate coupling matrix elements for mode-mode resonant interactions and show that the divergence appearing in previous studies can be eliminated completely by using a Fetter-like variational approximation for the ground state wave function of the condensate.We provide the selection rules in mode-mode interaction processes according to the symmetry of excitations.By solving the nonlinearly coupled envelope equations we obtain divergence-free nonlinear amplitudes for the three-mode resonant interaction and second harmonic generation processes and show that our theoretical results on the shape oscillations of the condensate agree well with the experimental ones.Finally,we discuss possible second-harmonic generation in a two-component BEC.