Stability of smooth periodic traveling waves in the Camassa–Holm equation

The candidate solution reproduces the paper’s central claims and methods: (i) the existence region bounded by three curves with ac=4c^3/27 and turning points given by (c−φ±)(φ±^2+2b)=2a; (ii) strict monotonicity of the period L in b for fixed a; (iii) for the nonstandard linearized operator K, exactly one simple negative eigenvalue, simple kernel spanned by μ′, and the rest of the spectrum strictly positive; and (iv) the fixed‑period continuation a↦BL(a), ΦL(·,a) and the stability criterion via the sign of d/da(EL/ML^2). These points align with Theorems 1.3–1.4 and supporting lemmas (e.g., Lemmas 2.2–2.4, 2.9, 3.2, 4.4, 4.6, 5.4–5.5) in the paper. Where the model mentions an alternative (KdV-normalization) path to monotonicity, the paper proves monotonicity by a planar Hamiltonian reduction; no substantive contradiction arises. Overall, the proofs and conclusions are essentially the same as in the paper. See Theorem 1.3 for the spectral count of K and period monotonicity, Theorem 2.6 for ∂bL>0, Lemma 2.9 for the fixed-period branch, and Lemma 4.6/Theorem 1.4 for the E/M^2 stability criterion .