Convective and Absolute Instabilities in the Subcritical Ginzburg-Landau Equation

TL;DR

This paper investigates the transition between convective and absolute instabilities in the subcritical Ginzburg-Landau equation, highlighting the effects of group velocity and nonlinearities through analytical and numerical methods.

Contribution

It provides a detailed analysis of how group velocity influences the absolute and convective instabilities in the subcritical Ginzburg-Landau equation, including the impact of finite size effects.

Findings

The absolute instability threshold is shifted by cubic nonlinearities.

Finite size effects can suppress the transition from convective to absolute instability.

High group velocities can make the middle branch of steady states convectively unstable.

Abstract

We study the nature of the instability of the homogeneous steady states of the subcritical Ginzburg-Landau equation in the presence of group velocity. The shift of the absolute instability threshold of the trivial steady state, induced by the destabilizing cubic nonlinearities, is confirmed by the numerical analysis of the evolution of its perturbations. It is also shown that the dynamics of these perturbations is such that finite size effects may suppress the transition from convective to absolute instability. Finally, we analyze the instability of the subcritical middle branch of steady states, and show, analytically and numerically, that this branch may be convectively unstable for sufficiently high values of the group velocity.