Standing waves of the complex Ginzburg-Landau equation

TL;DR

This paper proves the existence of nontrivial standing wave solutions for the complex Ginzburg-Landau equation under certain parameter conditions, expanding understanding of wave phenomena in nonlinear complex systems.

Contribution

It establishes the existence of standing wave solutions for the complex Ginzburg-Landau equation for a broad range of parameters with small nonlinearity exponent.

Findings

Existence of standing waves proven for all relevant parameter values.

Results cover all  and  cases where   .

Applicable for sufficiently small .

Abstract

We prove the existence of nontrivial standing wave solutions of the complex Ginzburg-Landau equation $\phi_t = e^{i\theta} \Delta \phi + e^{i\gamma} |\phi |^\alpha \phi $ with periodic boundary conditions. Our result includes all values of $\theta $ and $\gamma $ for which $\cos \theta \cos \gamma >0$, but requires that $\alpha >0$ be sufficiently small.