Stability of Traveling Oscillating Fronts in Complex Ginzburg Landau Equations

TL;DR

This paper proves the asymptotic stability of traveling oscillating fronts in complex Ginzburg-Landau equations, allowing for initial perturbations with localized and nonzero far-field components, using spectral and semigroup analysis.

Contribution

It establishes the stability of TOFs under broader initial conditions by analyzing the spectrum and employing an extended phase space approach.

Findings

Proves stability of TOFs with nonzero far-field limits.

Develops spectral analysis for operators with spectrum touching the imaginary axis.

Uses extended phase space and exponential weights for resolvent estimates.

Abstract

Traveling oscillating fronts (TOFs) are specific waves of the form $U_\star (x,t) = e^{-i \omega t} V_\star(x - ct)$ with a profile $V_{\star}$ which decays at $- \infty$ but approaches a nonzero limit at $+\infty$. TOFs usually appear in complex Ginzburg Landau equations of the type $U_t = \alpha U_{xx} + G(|U|^2)U$. In this paper we prove a theorem on the asymptotic stability of TOFs, where we allow the initial perturbation to be the sum of an exponentially localized part and a front-like part which approaches a small but nonzero limit at $+ \infty$. The underlying assumptions guarantee that the operator, obtained from linearizing about the TOF in a co-moving and co-rotating frame, has essential spectrum touching the imaginary axis in a quadratic fashion and that further isolated eigenvalues are bounded away from the imaginary axis. The basic idea of the proof is to consider the problem in an extended phase space which couples the wave dynamics on the real line to the ODE dynamics at $+ \infty$. Using slowly decaying exponential weights, the framework allows to derive appropriate resolvent estimates, semigroup techniques, and Gronwall estimates.