Stability of Plane Wave Solutions in Complex Ginzburg--Landau Equation with Delayed Feedback

We perform bifurcation analysis of plane wave solutions in a one-dimensional complex cubic-quintic Ginzburg--Landau equation with delayed feedback. Our study reveals how multistability and snaking behavior of plane waves emerge as time delay is introduced. For intermediate values of the delay, bifurcation diagrams are obtained by a combination of analytical and numerical methods. For large delays, using an asymptotic approach we classify plane wave solutions into strongly unstable, weakly unstable, and stable. The results of analytical bifurcation analysis are in agreement with those obtained by direct numerical integration of the model equation.