Periodic waves in the fractional modified Korteweg–de Vries equation

The paper’s main theorems on existence, Morse index, and co-periodic spectral stability for odd and even single-lobe periodic waves of D^αψ + cψ + b = 2ψ^3 (Theorems 1.3 and 1.6) are clearly stated and proved; the candidate solution reproduces these statements with a variational–index approach that is broadly consistent. In particular, it matches (i) existence via constrained minimization in parity subspaces, (ii) n(L)=2 for odd waves with a (typically) simple zero (if 1 ∈ Range(L)), and (iii) n(L)=1 for even waves, and it gives the same stability criteria in terms of σ0 = ⟨L^{-1}1,1⟩ and slope conditions d/dc||ψ||^2 or ∂_ω||φ||^2, in agreement with Theorem 2.13 and Theorem 4.8 of the paper . Two model-side issues require correction: (1) it asserts simplicity of the zero eigenvalue for odd waves and concludes 1 ∈ Range(L) by Fredholm, whereas the paper allows (and numerically observes) a parameter value where z(L)=2 and 1 ∉ Range(L) for the odd branch (stability bifurcation) ; and (2) its Birman–Schwinger/positivity-improving argument is insufficient to rule out more than one negative eigenvalue on the even subspace, although the desired bound does hold by a nodal/oscillation argument used in the paper. Apart from these correctable points, the candidate’s approach reaches the same conclusions as the paper.