Nonlinear structures of traveling waves in the cubic-quintic complex Ginzburg-Landau equation on a finite domain

We investigate the behavior of traveling waves in a spatial domain with the homogeneous boundary conditions by using the one-dimensional cubic-quintic complex Ginzburg-Landau equation. We focus our work on the absolute and convective instabilities and determine the dynamical regimes that are observed. As a consequence in the convectively unstable regime, the waves ultimately decay at any fixed position. Only when the threshold for the absolute instability is exceeded, the wave patterns may be maintained against the dissipation at the boundary. Consequently, coherent structures may be observed in the last case. We build a new state of phase diagram in the parameter plane spanned by the criticality parameter and the quintic nonlinear coefficient.