Numerical Continuation of Invariant Solutions of the Complex Ginzburg-Landau Equation

We consider the problem of computation and deformation of group orbits of solutions of the complex Ginzburg-Landau equation (CGLE) with cubic nonlinearity in $1\!+\!1$ space-time dimension invariant under the action of the three-dimensional Lie group of symmetries $A(x,t) \rightarrow \mathrm{e}^{\mathrm{i}\theta}A(x+\sigma,t+\tau)$. From an initial set of group orbits of invariant solutions, for a particular point in the parameter space of the CGLE, we obtain new sets of group orbits of invariant solutions via numerical continuation along paths in the moduli space. The computed solutions along the continuation paths are unstable, and have multiple modes and frequencies active in their spatial and temporal spectra, respectively. Structural changes in the moduli space resulting in symmetry gaining / breaking associated often with the spatial reflection symmetry $A(x,t) \rightarrow A(-x,t)$ of the CGLE were frequently uncovered in the parameter regions traversed.