Spatial structure of time-periodic solutions of the Ginzburg-Landau equation (Q1822265)

We study solutions of the form \(A(x,t)=a(x)e^{i\omega t}\) of the complex Ginzburg-Landau equation \(iA_ t+{\bar \alpha}A_{xx}={\bar \beta}A-{\bar \gamma}| A|^ 2A\), near those of the integrable limit: the nonlinear Schrödinger equation \(iA_ t+{\bar \alpha}A_{xx}=-{\bar \gamma}| A|^ 2A,{\bar \alpha},{\bar \gamma}\) real. We show that spatially periodic and quasiperiodic solutions as well as heteroclinic orbits to them exist, in addition to the spatially decaying (homoclinic) solutions found by \textit{L. M. Hocking} and \textit{K. Stewartson} [Proc. R. Soc. Lond., Ser. A 326, 289- 313 (1972; Zbl 0228.76074)]. We speculate briefly on spatially chaotic solutions.