ABSTRACT
This chapter outlines how exponential-precision asymptotics can be carried from ordinary differential equations to genuine partial differential equations. Exponential asymptotics was conceived for the purpose of tracing a simple, direct road to the prediction of physically observable quantities for which the asymptotic expansion is zero to all orders. The main purpose of all the simplifying assumptions is to cast the canonical equations into a concrete and explicit form from which many of the decisive qualitative facts can be read off lucidly. The symmetries of separable wave problems preclude such a distinction between ray envelopes and branch loci, but it is a normal feature of nonseparable wave problems and presents us with an ambiguity of the caustic concept. The ability to resolve Stokes phenomena on a strictly local level on individual trajectories is needed critically for nonlinear wave equations. Nonlinear wave equations open up so large a field that concrete, quantitative results can be anticipated only for specific sub-classes.
