BIFURCATION TO INSTABILITY THROUGH THE LENS OF THE MASLOV INDEX

The paper restates the known stability criterion for the standing single pulse of (2.1) and explains, via a Maslov-index calculation, how the sign of αV0 + (β/D)W0 governs stability; the statement and setup are explicit (system (2.1), linearization (3.1), essential spectrum left, and the criterion αV0 + (β/D)W0 < 0) and the jump-off values V0 = −e^{−2x*}, W0 = −e^{−2x*/D} are given to leading order . Their key crossing-form computation identifies the same sign combination (up to a minus sign), connecting the Maslov index to the stability threshold . The candidate solution arrives at the identical condition using a fast–slow Evans/Lin reduction near λ = 0, yielding a small “interaction” eigenvalue λ* ∼ C ε^2[αV0 + (β/D)W0] whose sign controls stability. Thus, both support the same criterion by different methods. The paper’s calculation is intentionally not a full spectral proof (it cites prior work to equate Maslov index with the count of unstable eigenvalues), whereas the model’s argument sketches that spectral route directly; taken together, they agree in result and are consistent with the literature.