Hysteretic behavior of stable solutions at the onset of a weakly inverted instability

We investigate in the framework of the quintic complex Ginzburg-Landau (CGL) equation in one spatial dimension the dynamics of the transition from moving pulse solutions to moving hole solutions, a new class of solutions found for this equation very recently. We find that the transition between these two classes of solutions is weakly hysteretic and that the velocity of moving pulses and moving holes shows a jump across the transition, that is moving particles and moving holes travel at different speeds on both sides of the transition.