Persistence of Periodic Traveling Waves and Abelian Integrals

The paper’s Theorem A proves persistence of periodic traveling waves under small perturbations via a near-Hamiltonian reduction and a Melnikov–Poincaré–Pontryagin integral M_c(h). The candidate solution reproduces the same reduction (Property A(i)), the Hamiltonian structure with weighted first integral H_c and time reparameterization s ↔ τ (Property A(ii)), and the Melnikov integral formula M_c(h)=∮_{γ_c(h)} g_c/s_c dx (Property A(iii)). It then invokes the standard result that simple zeros of M_c yield limit cycles and hence periodic TWS. This mirrors the paper’s proof almost step-for-step, differing only in presentation details (e.g., computing dH_c/ds directly versus explicitly switching to τ).