Existence of standing waves for the complex Ginzburg-Landau equation

TL;DR

This paper proves the existence of non-trivial standing wave solutions for the complex Ginzburg-Landau equation under certain conditions on parameters and domain, extending understanding of wave phenomena in nonlinear PDEs.

Contribution

It establishes the existence of standing waves for the complex Ginzburg-Landau equation in unbounded and bounded domains, with new conditions on parameters and domain geometry.

Findings

Existence of standing waves in $ n$ for specific parameter ranges.

Existence of standing waves in bounded domains with certain eigenvalue conditions.

Extension of previous results to more general settings and parameters.

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.