Dynamics near Couette flow for the β-plane equation

The candidate reproduces the right framework (traveling-wave reduction, Rayleigh–Kuo ODE, principal eigenvalue λ1(β,c), threshold αβ = √(−λ1(|β|,−1)), and the existence/nonexistence trichotomy) and many auxiliary properties match the paper. However, two critical points are missing or incorrect: (i) at the threshold T = Tβ (α = αβ), the corresponding eigenvalue is embedded in the continuous spectrum; the paper resolves this by bifurcating from a scaled shear (ay,0) to obtain an isolated eigenvalue and then passing to the limit, whereas the model attempts a direct Crandall–Rabinowitz argument at Couette without addressing the embedded-eigenvalue obstruction, which is not justified ; and (ii) in the “subcritical” regime and for T < Tβ, the model’s uniform implicit-function argument on nonzero x-modes does not control the case when the traveling speed c lies inside (−1,1), exactly where singular factors appear. The paper treats this regime via a separate nonexistence theorem that uses Hardy-type control (Theorem 1.1) . The rest of the model’s steps (derivation, spectral monotonicity and continuity, and the existence when α < αβ) agree with the paper’s logic and results .