The Eckhaus instability: from initial to final stages

TL;DR

This paper systematically analyzes the evolution of the Eckhaus instability in the one-dimensional Ginzburg-Landau equation through numerical simulations, identifying four distinct dynamic regimes from initial decay to final stabilization.

Contribution

It provides a detailed numerical study of the Eckhaus instability, characterizing the transition stages and the dynamics of the solution, spectrum, and Lyapunov functional.

Findings

Identification of four distinct regimes in the instability evolution.

Characterization of the phase-slip period with a sharp Lyapunov decrease.

Demonstration of slow relaxation to a stable state after instability.

Abstract

A systematic analysis of the Eckhaus instability in the one-dimensional Ginzburg-Landau equation is presented. The analysis is based on numerical integration of the equation in a large (xt)-domain. The initial conditions correspond to a stationary, unstable spatially periodic solution perturbed by "noise." The latter consists of a set of spatially periodic modes with small amplitudes and random phases. The evolution of the solution is examined by analyzing and comparing the dynamics of three key characteristics: the solution itself, its spatial spectrum, and the value of the Lyapunov functional. All calculations exhibit four distinct, mutually agreed, well-defined regimes: (i) rapid decay of stable perturbations; (ii) latent changes, when the solution and the Lyapunov functional undergo minimal alterations while the Fourier spectrum concentrates around the most unstable perturbations; (iii) a phase-slip period, characterized by a sharp decrease in the Lyapunov functional; (iv) slow relaxation to a final stable state.