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We consider the complex Ginzburg-Landau equation with two pure-power nonlinearities: $$ \partial_t u = (a + i\alpha) \Delta u + (b + i \beta) |u|^{\sigma_1} u - (c+i\gamma) |u|^{\sigma_2} +k u. $$ After proving a general global existence result, we focus on the existence and stability of several periodic orbits, namely the trivial equilibrium, bound-states and solutions independent of the spatial variable.

A generalized complex Ginzburg-Landau equation: Global existence and stability results