Hole solutions in the cubic complex Ginzburg-Landau equation versus holes in the cubic-quintic complex Ginzburg-Landau equation

In this contribution we compare the properties of hole solutions of the cubic complex Ginzburg-Landau equation with those of the cubic-quintic complex Ginzburg-Landau (CGL) equation in one spatial dimension. Both equations occur as prototype envelope equations near the onset of an oscillatory bifurcation to traveling waves. While hole solutions of the cubic CGL equation have been discussed already in detail for about two decades and are known to be structurally unstable, the study of stable hole solutions for the cubic-quintic CGL equation has only attracted more attention rather recently. The hole solutions of the latter equation turn out to be structurally stable. In addition, several classes of moving and breathing hole solutions have been found. Here we critically compare our current knowledge of hole solutions for both type of equations and also point out connections to hole solutions that have been found in other prototype equations describing pattern formation in dissipative systems.