The essential spectrum of periodically-stationary solutions of the complex Ginzburg-Landau equation

TL;DR

This paper analyzes the spectral properties of linearized operators around breather solutions of the complex Ginzburg-Landau equation, providing formulas for the essential spectrum and implications for pulse stability in fiber lasers.

Contribution

It introduces a method to determine the essential spectrum of the monodromy operator for periodically-stationary solutions of the complex Ginzburg-Landau equation, incorporating spectral filtering effects.

Findings

Derived a formula for the essential spectrum in terms of the asymptotic operator.

Established existence and regularity of the monodromy operator.

Discussed stability implications for ultrafast fiber laser pulses.

Abstract

We establish the existence and regularity properties of a monodromy operator for the linearization of the cubic-quintic complex Ginzburg-Landau equation about a periodically-stationary (breather) solution. We derive a formula for the essential spectrum of the monodromy operator in terms of that of the associated asymptotic linear differential operator. This result is obtained using the theory of analytic semigroups under the assumption that the Ginzburg-Landau equation includes a spectral filtering (diffusion) term. We discuss applications to the stability of periodically-stationary pulses in ultrafast fiber lasers.