Diffusive Stability of Convective Turing Patterns

The paper proves, via a direct Lyapunov–Schmidt reduction of the Bloch operator and a three-regime Bloch-number analysis, that Eckhaus’ criterion exactly characterizes diffusive spectral stability for small-amplitude convective Turing patterns (Theorem 1.2), and it matches the complex Ginzburg–Landau (cGL) spectral expansion up to O(ε) in the coefficients (Theorem 1.3) . The candidate solution argues instead by invoking standard center-manifold/modulation theory to derive cGL and transfer its Eckhaus result back to the original operators, reaching the same stability/instability conclusions. While the model omits some subtleties (e.g., the O(σ) purely imaginary drift and the affine Bloch-frame shift seen in the convective case), its conclusions on Re λ and the diffusive bound align with the paper’s results. Hence both are correct, but the proofs are methodologically different.