The Eckhaus instability: From initial to final stages

A systematic analysis of the Eckhaus instability in the one-dimensional real Ginzburg–Landau equation is presented. The analysis is based on numerical integration of the equation in a large ( x t )-domain. The initial conditions correspond to a stationary, unstable spatially periodic solution perturbed by “noise”. The latter consists of a large discrete 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. The findings appear to be general and offer fresh insights into this longstanding and significant issue. • The evolution from an unstable to a stable stationary spatially periodic solution of the real GL equation initiated by a broad-band, small amplitude perturbation proceeds through four sequential, qualitatively distinct stages. • Each stage is thoroughly described, its physical basis established, and its distinctive dynamics explained. • The findings admit experimental verification.