Convective Turing Bifurcation

The candidate’s Lyapunov–Schmidt reduction, decomposition into the critical Fourier modes ℓ = ±1 and their complement, slaving of mean and second-harmonic corrections, and projection onto adjoint eigenvectors to obtain the complex cubic normal form agree with the paper’s P/Q/R projector-based Lyapunov–Schmidt scheme and the complex Ginzburg–Landau (cGL) plane-wave dispersion relation. The existence (inside the Eckhaus band) and nonexistence (outside, with fixed margin) statements, and the O(2) symmetric stationary case, match Theorem 1.2 and the formulas (1.10)–(1.14) in the paper. Minor differences are notational (e.g., sign/normalization in the second-harmonic resolvent) and a small scaling slip in the candidate’s use of ε² in the detuning bound; the logical structure and conclusions coincide. See the paper’s plane-wave dispersion relation and band criterion (1.10)–(1.13) and Theorem 1.2 for the existence/nonexistence and O(2) claims, and the Lyapunov–Schmidt setup and invertibility on the complement (4.17)–(4.22) with X = Ψ(V,W) (4.21). The derivation of cGL from multiscale expansion and identification of γ also agree, including γCGL = γLS (Corollary 5.16).