Strict and fussy mode splitting in the tangent space of the Ginzburg-Landau equation

TL;DR

This paper investigates the mode splitting in the tangent space of the Ginzburg-Landau equation, revealing how physical and isolated modes behave under different spatial coupling strengths and the transition from strict to fuzzy splitting.

Contribution

It provides a detailed analysis of mode splitting phenomena in the Ginzburg-Landau equation, highlighting the conditions under which strict and fuzzy mode splitting occur.

Findings

Isolated modes match eigenmodes of the steady state.

A local basis can be constructed where non-zero components match physical modes.

Strict mode splitting disappears at finite coupling, leading to fuzzy splitting.

Abstract

In the tangent space of some spatially extended dissipative systems one can observe "physical" modes which are highly involved in the dynamics and are decoupled from the remaining set of hyperbolically "isolated" degrees of freedom representing strongly decaying perturbations. This mode splitting is studied for the Ginzburg-Landau equation at different strength of the spatial coupling. We observe that isolated modes coincide with eigenmodes of the homogeneous steady state of the system; that there is a local basis where the number of non-zero components of the state vector coincides with the number of "physical" modes; that in a system with finite number of degrees of freedom the strict mode splitting disappears at finite value of coupling; that above this value a fussy mode splitting is observed.