Damped nonlinear Ginzburg-Landau equation with saturation. Part I. Existence of solutions on general domains

TL;DR

This paper investigates the complex Ginzburg-Landau equation with saturated nonlinear damping on unbounded domains, establishing existence and uniqueness of solutions through advanced operator theory.

Contribution

It extends the analysis of the Ginzburg-Landau equation to include saturated nonlinear damping and unbounded domains, providing new existence and uniqueness results.

Findings

Proved existence of global solutions under certain conditions.

Demonstrated the applicability of maximal monotone operator theory to this framework.

Handled complex coefficients and unbounded domains effectively.

Abstract

We study the complex Ginzburg-Landau equation posed on possibly unbounded domains, including some singular and saturated nonlinear damping terms. This model interpolates between the nonlinear Schr{\"o}dinger equation and dissipative parabolic dynamics through a complex time-derivative prefactor, capturing the interplay between dispersion and dissipation. Under suitable structural conditions on the complex coefficients, we establish the existence and uniqueness of global solutions. The analysis relies on the delicate proofs that the maximal monotone operator theory can be adapted to this framework, even for unbounded domains.