Exact periodic solutions of the complex Ginzburg–Landau equation

Three new exact periodic solutions of the complex Ginzburg–Landau equation are obtained in terms of the Weierstrass elliptic function ℘. Furthermore, the new periodic solutions and other shock solutions appear as their bounded limits (along the real axis) for particular relationships between the coefficients in the equation. In the general case, bounded limits are nothing but the already known pulse, hole, and shock solutions. It is also shown that the shapes of the solutions are quite different from the shape of the usual envelope wave solution. In particular, the spatial structure of the new bounded periodic solutions varies in time, while the pulse solution may exhibit breather-like behavior.