Stability of Smooth Periodic Traveling Waves in the Degasperis–Procesi Equation

The paper formulates DP as a Hamiltonian PDE du/dt = J δF/δu with J = −(1−∂x^2)^{-1}(4−∂x^2)∂x and F(u)=∮u^3/6, introduces the augmented energy Λc,b and its Hessian L = c − φ − 3c(4−∂x^2)^{-1} at a smooth L‑periodic travelling wave φ, and linearizes to wt = −JLw; it then enforces the co‑periodic constraints 〈1,w〉=0, 〈φ^2,w〉=0 (invariance shown using J1=0 and Jφ^2=−2cφ′, with Lφ′=0) to reduce spectral stability to the quadratic form of L on X0 and a 2×2 projection matrix S with entries 〈L−1·,·〉. Along the fixed‑period branch a↦b=BL(a), det S is expressed purely via the monotonicity of a↦FL/ML^3 (and a↦ML when B′L<0), yielding n(L|X0)=0 and z(L|X0)=1, hence Spec(JL)⊂iℝ; these conditions are proved near the constant‑wave boundary via a Stokes expansion (Lemma 5.6) and observed numerically in the interior (Theorem 1.2). All these steps appear explicitly in the paper, including (i) the Hamiltonian structure and Hessian (1.5)–(1.8) , (ii) the linearized form and constraints (5.1)–(5.3) and X0 definition (5.2) , (iii) the constrained index count n(L|X0)=n(L)−n0−z0 via the projection matrix S (5.6)–(5.7) and its evaluation leading to det S∝d/dβ( F̂L/M̂L^3 ) (5.9)–(5.11) , (iv) the split by the sign of B′L and the need for monotonicity of ML when B′L<0 (5.12) , and (v) the small‑amplitude verification near a=a−(b) (Lemma 5.6) . The candidate solution reproduces precisely this constrained‑energy/Hamiltonian–Krein scheme, including the same J, the same Hessian L, the same two constraints, the same S‑matrix reduction via parameter derivatives, and the same monotonicity criterion. Minor issues in the candidate write‑up include a momentary ambiguity about the second constraint (it is 〈φ^2,w〉, not 〈φ,w〉) and the need to exclude tangencies (B′L(a)=0) and the special interior curve where L has a double zero, both of which the paper treats explicitly. Net: the arguments and conclusions agree, and the proof strategy is substantially the same.