Proof of the Absence of Elliptic Solutions of the Cubic Complex Ginzburg–Landau Equation

We consider the cubic complex Ginzburg–Landau equation. Using Hone's method, based on formal Laurent-series solutions and the residue theorem, we prove the absence of elliptic standing-wave solutions of this equation. This result complements a result by Hone, who proved the nonexistence of elliptic traveling-wave solutions. We show that it is more efficient to apply Hone's method to a system of polynomial differential equations rather than to an equivalent differential equation.