Dynamics of matter-wave condensates with time-dependent two-and three-body interactions trapped by a linear potential in the presence of atom gain or loss

Bose-Einstein condensates with time varying two-and three-body interatomic interactions, confined in a linear potential and exchanging atoms with the thermal cloud are investigated. Using the extended tanh-function method with an auxiliary equation, i.e., the Lenard equation, many exact solutions describing the dynamics of matter-wave condensates are derived. An important issue is the time management of the cubic and the quintic nonlinearities by tuning the rate of exchange of atoms between the condensate and the thermal background. In addition, adjusting the strength of the linear potential, the rate of exchange of atoms, and many other free parameters allow one to control many features of the condensate such as its height, width, position, velocity, acceleration, and its direction, respectively. Full numerical solutions corroborate the analytical predictions.