Interacting chirped localized waves in Bose-Einstein condensates with time-varying complex potentials

We examine the interaction of chirp localized matter waves in a cubic-quintic Gross-Pitaevskii equation with variable coefficients (model equation) that governs the dynamics of Bose-Einstein condensates in a time-varying complex potential, which is composed of a parabolic background potential and a gravitational field when the atomic loss/gain is taken into account. By performing a phase imprint transformation, we derive the integrability conditions for the model equation. Under the integrability conditions, the model equation is converted into a nonlinear Schrödinger equation with self-steepening (alias similarity equation). An explicit expression for the growth rate of a purely growing baseband modulational instability is derived for the similarity equation. The criterion of the baseband modulational instability is found to correspond to Bose-Einstein condensates with attractive two-body interatomic interactions. We shown that the gain/loss parameter does not affect the baseband modulational instability. Employing the generalized perturbation (n, N − n)−fold Darboux transformation, new mixed localized wave solutions with nonlinear frequency chirp are built for the model equation. Those solutions helping, we analyze graphically the formation and interaction of chirp localized matter waves in Bose-Einstein condensates with both two-and three-body interactions when the gain/loss of the condensate atoms is taken into consideration. Our results show 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 similarity parameters are useful for controlling the chirped matter localized waves for the model under consideration when the integrability conditions are satisfied. Focusing on the situation when the parameter of the three-body interaction is proportional to that of the two-body interaction, we show that parameter of the two-body interatomic interaction can be used to describe wave compression and also, can be used for the manipulation of chirped localized waves in Bose-Einstein condensates in complicated potential. The spectral parameter is found to be useful for for modeling localized chirped wave structure 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 (the wave amplitude remains constant during the wave propagation).