Adiabatic Asymptotics of the Amsler Equation Solution in Closed Form

We consider the regular solution \(y(x;s)\) of the Amsler equation \((xy')'=\sin y\), \(y(0)=\pi s\), \(y'(0)=\sin\pi s\), with \(s\in(0,1)\), and prove that as \(x\to\infty\) it is approximated, with accuracy \(O(\tau^{-5/2})\) for \(\tau=2\sqrt{x}\), by an explicit adiabatic ansatz \(y_{\rm ad}\): a libration of the pendulum \(w''+\sin w=0\) (\(w=y-\pi\)) whose parameter \(m_0(\tau)\) is determined by the adiabatic invariant \(\tau\,\ell(m_0)=\pi\nu\), where \(\ell(m)=E(m)-(1-m)K(m)\) and \(\nu=\pi^{-1}\ln\csc\frac{\pi s}{2}\), and whose phase is \(\Phi=\tau f(m_0)+\Theta_0-\frac{1}{8\tau}\) with \(f(m)=E(1-m)-mK(1-m)\) and \(\Theta_0=C_2+\nu(1+\ln\frac{8}{\nu})\). The proof combines elementary adiabatic analysis of the damped pendulum (residual estimates, absence of low-order resonances, an approximate fundamental system) with two external inputs stated explicitly: Maevsky's asymptotics for \(y\) and the Its--Novokshenov/Kitaev connection formula for the degenerate Painleve III equation, which determine the amplitude and phase constants in closed form, \(C_1=\sqrt{8\nu}\), \(C_2=\frac{5\pi}{4}+\arg\Gamma(1+i\nu)-3\nu\ln2\).