Existence of standing waves for the complex Ginzburg-Landau equation

Abstract

We prove the existence of non-trivial standing wave solutions of the complex Ginzburg-Landau equation $\phi_t - e^{i\theta}(\rho I- \Delta) \varphi - e^{i\gamma} |\phi |^\alpha \varphi =0 $ in $\Rn$, where $(N-2)\alpha <4$, $\theta ,\gamma \in (-\pi /2,\pi /2)$ and $\rho >0$. Analogous result is obtained in a ball $\Omega \in\Rn$ for $\rho >-\lambda_1$, where $\lambda_1$ is the first eigenvalue of the Laplace operator with Dirichlet boundary conditions.