A cubic-quintic Gross-Pitaevskii equation (model equation) that governs the dynamics of Bose-Einstein condensates in a time-varying complex potential is considered for examining interacting chirped localized matter waves in Bose-Einstein condensates with a time-varying potential when the atomic loss/gain is taken consideration. After presenting the integrability conditions, we establish the criterion of the baseband modulational instability. Based on the generalized perturbation ( \(n,N-n\) )−fold Darboux transformation, we present new mixed localized wave solutions with nonlinear frequency chirp. With the aid of these solutions, we investigate interacting chirped localized matter waves in the physical system under consideration. Our results show a riche variety of interactions between various kinds of nonlinear waves such as multi-peak bright/dark solitons, bright/dark breathers, as well as periodic waves. We show that the parameter of the interatomic interactions can be used for describing wave compression and for manipulating chirped localized waves in Bose-Einstein condensates in complicated potential. Also, we show that the spectral parameter is useful for modeling chirped localized wave structures in the model under consideration. Interestingly, we show that under the integrability conditions, the gain/loss parameter does not affect the wave amplitude during wave propagation.
