The Maslov index, degenerate crossings and the stability of pulse solutions to the Swift-Hohenberg equation

The paper’s Theorem 1.5 precisely asserts that, under Hypothesis 3.17, the number of unstable eigenvalues of the linearized Swift–Hohenberg operator equals the number of conjugate points for the path ℓ(x)=Eu−(x;0) relative to the sandwich plane ℓsand∗, and it proves this via a Maslov-box argument on the half-line, together with a careful treatment of degenerate crossings using higher-order crossing forms (notably third order in this problem) and a continuation to the whole line (Theorems 3.1 and 3.20) . By contrast, the candidate solution misinterprets Hypothesis 3.17 as ruling out degenerate crossings (it does not), assumes all crossings are regular and uses a first-order crossing form positivity claim to conclude that the Maslov index equals the unsigned count of crossings. The paper explicitly shows that simple but nonregular (degenerate) crossings occur and must be handled with higher-order crossing forms: for vectors v ∈ ℓ(s∗,λ)∩ℓsand∗, one can have Q(1)(v)=0 while Q(3)(v)>0, which still yields the needed monotonicity and correct signed contributions (Lemmas 3.13 and 3.16; Proposition 3.18) . The candidate also asserts a bounded-interval identity with sandwich conditions at both endpoints without establishing it for this fourth-order setting, whereas the paper proves the count on a half-line with sandwich boundary at the right endpoint and then passes to the full line (Definition 3.4, Lemma 3.5, and Theorem 3.20) . Hence the paper’s argument is complete under its hypotheses; the candidate’s argument omits the essential degenerate-crossing analysis and misstates the hypothesis’ implications.