Asymptotics of solutions to a second-order quasi-linear equation (Q1571264)

The author examines the asymptotic behavior of the real solution \(z(t)\) to the equation (1) \(tz''+z'+tf(z)=0\), where \(f(\xi)\) is a real function subjected to certain constraints. If \(f(\xi)=\sin{\xi}\), the equation is written as (2) \(tz''+z'+t\sin{z}=0\). This case is especially interesting since the well-known sine-Gordon equation reduces to equation (2). The asymptotics of solutions are obtained in case when \(f\) negligibly differs from sine, and the solutions satisfy initial conditions at \(t\in{\mathbb R_+}\) that are sufficiently small in absolute value.