Floquet stability of periodically stationary pulses in a short-pulse fiber laser

The paper’s Proposition 6.1 proves that the modified monodromy operator M̃ has two unit eigenfunctions, the phase mode Jψ and the translation mode ψx, implying a unit Floquet multiplier with multiplicity at least two. The candidate solution derives exactly the same identities by differentiating the phase- and translation-equivariance and then precomposing with R(−θ). It also supplies a clear linear-independence argument for {Jψ, ψx}, which the paper asserts implicitly. No substantive conflict exists; the proofs are essentially the same.