Why is a shock not a caustic? The higher-order Stokes phenomenon and smoothed shock formation

Abstract

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.

Exponentially accurate solution tracking for nonlinear ODEs, the higher order Stokes phenomenon and double transseries resummation

“…(66)). As in [7] presence of the higher order Stokes curve |z| = 1 has forced the singularityf 2 to move onto a different Riemann sheet from the point f 0 . Therefore there is no coalescence of these singularities even though f 2 (0) = f 0 (0).…”

“…Virtual turning points have been shown [7] to be associated with the presence of "smoothed shocks", where there is a rapid (monotonic) local transition between different values of the solution. All of these effects can be studied by considering the related singularity structure in the Borel plane representation of the problem.…”

“…Techniques illustrated in [2], [7] and [1] show that, in principle, it is possible to identify and follow such terms asymptotically. However a novel approach is required to seed and track this numerically.…”

“…In [16] the effect of the higher order Stokes phenomenon in an inhomogeneous second order linear ordinary differential equation was illustrated. In [7] we showed in the context of Burgers equation how the presence of a higher order Stokes phenomenon prevents the presence of singularities in uniform asymptotic approximations of the solution after the formation of a nonlinear shock.…”

“…The incorporation of a single exponentially small term by resumming it into a multiple-scales approach is a useful way of extending the range of validity of the solution as the dominance of the terms change, especially in nonlinear contexts [7], [17], [15]. However such uniform approaches are relatively unexplored when a third, initially sub-subdominant exponential, is also present.…”