The diversity of steady state solutions of the complex Ginzburg-Landau equation

The structure of the phase space of stationary and quasi-stationary (i.e., uniformly translating) solutions of 1D CGLE is investigated by methods of the qualitative theory of ordinary differential equations. The Nozaki-Bekki holes are seen as heteroclinic connections which are made structurally stable by an involution symmetry in phase space, The existence of a countable set of double-loop heteroclinic trajectories is proved, which corresponds to complex ''shock-hole-shock'' structures both motionless and moving with constant velocity v(0) along the x-axis.