An Amsler Profile Periodic Function: Asymptotic Series from Nonlinear Oscillations in the Amsler Equation

This document presents a rigorous analysis of the Amsler Profile PeriodicFunction Pε(u;C1), which emerges from the asymptotic expansion of solutions of the Amsler equation tw′′+w′ = sinw. The function is defined as a formal asymptoticseries in a small parameter ε = 1/(2√t) and encodes the shape of the oscillatoryprofile at any fixed time scale. We prove that Pε is 2π-periodic in its argument,that its series provides an asymptotic expansion of the normalized amplitude whenu is taken as the phase θ(t), and that its coeffcients are uniquely determined by therecurrence derived from the Amsler equation. A corrected multiple-scale analysis iscarried out step by step, yielding the correct negative logarithmic phase shift anda Duuffng-type equation satisfied by the leading term. Fourier coeffcients up toorder ε2 are computed explicitly and compared with the recurrence. Connections toBessel functions and elliptic functions are discussed. All results are consistent withthe rigorous asymptotic theory of the Amsler equation (see [3]).