Traveling waves in the complex Ginzburg-Landau equation (Q1322784)

Travelling wave solutions of the one-dimensional Ginzburg-Landau equation \[ u_ t = u + (1 + i \alpha) u_{xx} - (1 + i \beta) | u |^ 2u, \quad x \in \mathbb{R} \tag{1} \] are studied. The equation is assumed to be weakly complex, that is, \(\alpha = \varepsilon a\), \(\beta = \varepsilon b\) with \(0<\varepsilon \ll 1\). For \(\varepsilon = 0\), the equation for travelling waves reduces to a 3-dimensional integrable system of ODEs. Using this fact, the author constructs certain travelling waves of (1) by continuation from the ``real case'' \(\varepsilon = 0\). In particular, two-parameter families of spatially quasiperiodic solutions are obtained this way.