Existence of Standing Waves for the Complex Ginzburg–Landau Equation

We study the existence of standing wave solutions of the complex Ginzburg–Landau equation(GL)φt−eiθ(ρI−Δ)φ−eiγ|φ|αφ=0 in RN, where α>0, (N−2)α<4, ρ>0 and θ,γ∈R. We show that for any θ∈(−π/2,π/2) there exists ε>0 such that (GL) has a non-trivial standing wave solution if |γ−θ|<ε. Analogous result is obtained in a ball Ω∈RN for ρ>−λ1, where λ1 is the first eigenvalue of the Laplace operator with Dirichlet boundary conditions.