PERIODIC WAVES OF THE MODIFIED KDV EQUATION AS MINIMIZERS OF A NEW VARIATIONAL PROBLEM

The paper sets up the focusing mKdV traveling-wave equation −ψ″+cψ+b=2ψ^3 together with the conserved functionals E, F, M and the action G_{c,b}=E+cF+bM, whose Hessian at a wave profile is L=−∂_x^2+c−6ψ^2 . For non-degenerate minimizers (positivity of L on {1,ψ^3}^⊥ with a simple kernel spanned by ψ′), the authors derive stability/instability criteria via resolvent-based matrices P(λ) and D(λ) and index-counting formulas; crucially, they show lim_{λ→0} det D(λ) = (1/∂_m b)·det(∂(F,M)/∂(c,m)) (up to the factor arising from F=½∮ψ^2), and n(L|_{ {1,ψ}^⊥}) = n(L) − n_0 − z_0 with cases partitioned by the sign of ∂_m b, yielding the sufficient stability criterion det(∂(F,M)/∂(c,m))>0 for minimizers and the instability criterion ∂_m b<0 together with the same Jacobian sign for saddle points . The candidate solution reproduces the parametric identities L∂_cψ = −ψ − (∂_c b)·1, L∂_mψ = −(∂_m b)·1, and ker L = span{ψ′} correctly and it recovers det G = J/∂_m b for G_{ij}=⟨v_i,L^{-1}v_j⟩ with v_1=1, v_2=ψ (consistent with D(0) in the paper) . However, its core coercivity/index argument is flawed: (i) it asserts “G is negative definite” in the stable case from positivity of L on W, but the paper’s stable case with ∂_m b<0 necessarily yields det G<0 (hence G is indefinite, not negative definite) when the Jacobian is positive ; (ii) it uses an incorrect “additive” index formula n_−(Q|_{T_ψΣ∩ψ′⊥}) = n_−(L|_W) + n_−(G), contradicting the paper’s established reduction n(L|_{ {1,ψ}^⊥}) = n(L) − n_0 − z_0 based on D(λ) . These sign/index inconsistencies undermine the model’s claimed coercivity and instability deductions, even though its high-level conditions (J>0 for stability; J>0 and ∂_m b<0 for instability) match the paper’s final criteria.