Standing waves of the complex Ginzburg-Landau equation

We prove the existence of nontrivial standing wave solutions of the complex Ginzburg-Landau equation $\phi_t = e^{i\theta} \Delta \phi + e^{i\gamma} |\phi |^\alpha \phi$ with periodic boundary conditions. Our result includes all values of $\theta$ and $\gamma$ for which $\cos \theta \cos \gamma >0$, but requires that $\alpha >0$ be sufficiently small.