Exponential asymptotics of woodpile chain nanoptera using numerical analytic continuation

The paper derives a general exponentially small tail formula for the woodpile–Toda nanopteron, v_exp ~ (2π|V|/η^ν) exp(−√k Im ξ_s/(cη)) cos(√k(ξ−Re ξ_s)/(cη) − φ) with V and φ determined by the local singularity type (logarithmic ⇒ V=μ, ν=0, φ=π/2) and identifies the two nearest logarithmic singularity pairs at ξ = ±1/2 ± iπ/(2κ). Summing the two dominant contributions yields the envelope Amplitude(η) ~ 4π exp(−√k π/(2κ c η)) sin(√k/(2cη)) (their eq. (25)), and thus zeros at √k/(2cη)=mπ. The candidate solution reproduces these results step-by-step: same traveling-wave reduction and leading-order Toda solitary wave, same singularity structure and late-term ansatz, same singulant magnitude and WKB remainder, same Stokes switching, and the same final amplitude and zero set. Minor differences (a sign convention in the singulant equation and an unnecessary assumption A=B for late terms where the paper shows v_j dominates u_j) do not affect the final outcomes. Therefore both are correct and essentially use the same exponential-asymptotics proof framework, culminating in the same amplitude and cancellation law. See the paper’s equations (20), (22)–(25), and Appendix A (late-order and singulant derivation) for precise statements and derivations .