Why is a shock not a caustic? The higher-order Stokes phenomenon and smoothed shock formation - Chapman, S; Howls, C; King, JR; Olde Daalhuis, AB (2007)
The formation of shocks in waves of advance in nonlinear partial differential equations is a well-explored problem and has been studied using many different techniques. In this paper we demonstrate how an exponential-asymptotic approach can be used to completely characterize the shock formation in a nonlinear partial differential equation and so resolve an apparent paradox concerning the asymptotic modelling of shock formation. In so doing, we find that the recently discovered higher-order Stokes phenomenon plays a significant, previously unrealized, role in the asymptotic analysis of shocks. For the purposes of clarity, Burgers' equation is used as a pedagogical example, but the techniques illustrated are more generally applicable. © 2007 IOP Publishing Ltd and London Mathematical Society.