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 ${\varphi }_t-e^{i\theta }(\rho I-\Delta )\phi -e^{i\gamma }|\varphi {|}^{\alpha }\phi =0$ in $\Rn$, where $(N-2)\alpha <4$, $\theta ,\gamma \in (-\pi /2,\pi /2)$ and $\rho >0$. Analogous result is obtained in a ball $Ω\in\Rn$ for $\rho >-{\lambda }_1$, where ${\lambda }_1$ is the first eigenvalue of the Laplace operator with Dirichlet boundary conditions.

References in corpus (1)

• From the defocusing nonlinear Schroedinger to the complex Ginzburg-Landau equation