Three-term spectral asymptotics for the perturbed simple pendulum problems

Abstract

This paper is concerned with the perturbed simple pendulum problem $$ -u''(t) + g(u(t)) = \lambda \sin u(t), \ \ u(t) > 0, \ \ t \in I := (-T, T), \ \ u(\pm T) = 0, $$ where $\lambda > 0$ is a parameter. It is known that if $\lambda \gg 1$, then the corresponding solution develops the boundary layers. We adopt a new parameter $\epsilon \in (0, T)$ which characterizes both the boundary layers and the height of the solution and parametrize a solution pair $(\lambda, u)$ by $\epsilon$, namely, $(\lambda, u) = (\lambda(\epsilon), u_\epsilon),$ and establish the three-term asymptotics for $\lambda(\epsilon)$ as $\epsilon \to 0$.