Moving and colliding pulses in the subcritical Ginzburg-Landau model   with a standing-wave drive
  We show the existence of steadily moving solitary pulses (SPs) in the complex Ginzburg-Landau (CGL) equation, which includes the cubic-quintic (CQ) nonlinearity and a conservative linear driving term, whose amplitude is a standing wave with wavenumber $k$ and frequency $\omega $, the motion of the SPs being possible at velocities $\pm \omega /k$, which provide locking to the drive. A realization of the model may be provided by traveling-wave convection in a narrow channel with a standing wave excited in its bottom (or on the surface). An analytical approximation is developed, based on an effective equation of motion for the SP coordinate. Direct simulations demonstrate that the effective equation accurately predicts characteristics of the driven motion of pulses, such as a threshold value of the drive's amplitude. Collisions between two solitons traveling in opposite directions are studied by means of direct simulations, which reveal that they restore their original shapes and velocity after the collision. 
B. B. Baizakov, G. Filatrella, and B. A. Malomed