Nonlinear stability of source defects in the complex Ginzburg-Landau equation

TL;DR

This paper proves the nonlinear stability of source defects in the complex Ginzburg-Landau equation by analyzing linearized dynamics, approximating solutions with Burgers equation, and deriving detailed resolvent and Green's function estimates.

Contribution

It provides a rigorous proof of nonlinear stability for source defects in the complex Ginzburg-Landau equation, including detailed linear and nonlinear analysis techniques.

Findings

Established nonlinear stability of source defects.

Developed an approximation using the nonlinear Burgers equation.

Derived detailed resolvent kernel and Green's function estimates.

Abstract

In an appropriate moving coordinate frame, source defects are time-periodic solutions to reaction-diffusion equations that are spatially asymptotic to spatially periodic wave trains whose group velocities point away from the core of the defect. In this paper, we rigorously establish nonlinear stability of spectrally stable source defects in the complex Ginzburg-Landau equation. Due to the outward transport at the far field, localized perturbations may lead to a highly non-localized response even on the linear level. To overcome this, we first investigate in detail the dynamics of the solution to the linearized equation. This allows us to determine an approximate solution that satisfies the full equation up to and including quadratic terms in the nonlinearity. This approximation utilizes the fact that the non-localized phase response, resulting from the embedded zero eigenvalues, can be captured, to leading order, by the nonlinear Burgers equation. The analysis is completed by obtaining detailed estimates for the resolvent kernel and pointwise estimates for the Green's function, which allow one to close a nonlinear iteration scheme.