On the Cauchy problem for the 1+2 complex Ginzburg-Landau equation

Abstract

We present analytical methods to investigate the Cauchy problem for the complex Ginzburg-Landau equation u₁ = (v + iα)Δu − (κ + iβ) |u|^2qu + γu in 2 spatial dimensions (here all parameters are real). We first obtain the local existence for v > 0, κ ≥ 0. Global existence is established in the critical case q = 1. In addition, we prove the global existence when .