Stokes-multiplier expansion in an inhomogeneous differential equation with a small parameter

Abstract

Accurate approximations to the solutions of a second-order inhomogeneous equation with a small parameter ϵ are derived using exponential asymptotics. The subdominant homogeneous solutions that are switched on by an inhomogeneous solution through a Stokes phenomenon are computed. The computation relies on a resurgence relation, and it provides the ϵ -dependent Stokes multiplier in the form of a power series. The ϵ -dependence of the Stokes multiplier is related to constants of integration that can be chosen arbitrarily in the WKB-type construction of the homogeneous solution. The equation under study governs the evolution of special solutions of the Boussinesq equations for rapidly rotating, strongly stratified fluids. In this context, the switching on of subdominant homogeneous solutions is interpreted as the generation of exponentially small inertia–gravity waves.