Criteria for the (in)stability of planar interfaces in singularly perturbed 2-component reaction-diffusion equations

Both the paper and the candidate solution use the same adjoint–projection strategy around the translational eigenvalue, split the adjoint eigenfunction into fast/slow pieces using the slow–fast–slow geometry, and deduce the sign rules for the long-wave transverse eigenvalue coefficient. The paper gives a clear derivation of the universal Rayleigh-type quotient (2.14), the fast/slow matching relation (2.19), and the leading-order expression (2.22) that implies sign(λ2,c) = −sign(F*) sign(G*) for τ = O(1) and, in the τ = ετ̃ regime, the formula analogous to (3.5)–(3.7) that yields sign(λ̃2,c) = −sign(F̃*) sign(M̃*) and the explicit coefficient (3.11). The candidate solution reaches these same sign criteria and the τ = ε explicit formula. However, the candidate’s writeup contains minor but nonfatal inconsistencies: missing τ and G* factors in the amplitude matching, an O(ε) scaling omission for the slow adjoint component when τ = O(1), and an unnecessary (and incorrect) statement about sign(D). These do not affect the final signs or the τ = ε explicit formula, since the common amplitude cancels in the Rayleigh quotient and the leading-order balances match the paper’s. Overall, both are correct and follow substantially the same proof idea; the paper is a careful sketch with explicit formulas, while the model solution needs minor corrections in intermediate steps.