A singularly perturbed linear partial differential equation motivated by the geometrical model for crystal growth is considered. A steepest descent analysis of the Fourier transform solution identifies asymptotic contributions from saddle points, end points and poles, and the Stokes lines across which these may be switched on and off. These results are then derived directly from the equation by optimally truncating the naïve perturbation expansion and smoothing the Stokes discontinuities. The analysis reveals two new types of Stokes switching: a higher-order Stokes line which is a Stokes line in the approximation of the late terms of the asymptotic series, and which switches on or off Stokes lines themselves; and a second-generation Stokes line, in which a subdominant exponential switched on at a primary Stokes line is itself responsible for switching on another smaller exponential. The ‘new’ Stokes lines discussed by Berk
et al
. (Berk
et al
. 1982
J. Math. Phys.
23
, 988–1002) are second-generation Stokes lines, while the ‘vanishing’ Stokes lines discussed by Aoki
et al
. (Aoki
et al
. 1998 In
Microlocal analysis and complex Fourier analysis
(ed. K. F. T. Kawai), pp. 165–176) are switched off by a higher-order Stokes line.
An elementary representation of the higher-order Jacobi-type differential equation
