Exponential asymptotics and boundary-value problems: keeping both sides happy at all orders

Abstract

We introduce templates for exponential-asymptotic expansions that, in contrast to matched asymptotic approaches, enable the simultaneous satisfaction of both boundary values in classes of linear and nonlinear equations that are singularly perturbed with an asymptotic parameter e → 0 + and have a single boundary layer at one end of the interval. For linear equations, the template is a transseries that takes the form of a sliding ladder of exponential scales. For nonlinear equations, the transseries template is a two-dimensional array of exponential scales that tilts and realigns asymptotic balances as the interval is traversed. An exponential-asymptotic approach also reveals how boundary-value problems force the surprising presence of transseries in the linear case and negative powers of e terms in the series beyond all orders in the nonlinear case. We also demonstrate how these transseries can be resummed to generate multiplescales-type approximations that can generate uniformly better approximations to the exact solution out to larger values of the perturbation parameter. Finally, we show for a specific example how a reordering of the terms in the exponential asymptotics can lead to an acceleration of the accuracy of a truncated expansion.

Resurgent aspects of applied exponential asymptotics

“…Indeed the work of Howls [14] finds a similar structure. Here, we mostly avoid this issue by always assuming that the Borel transform, with the perturbative contour γ = (0, ∞), satisfies the boundary condition.…”

“…There are a number of related works by e.g. Howls [14] (on trans-series for boundary-value problems) and Byatt-Smith [15] (on the Borel transform), but we believe the approach of working entirely in the Borel plane for such singularly perturbed problems is not as well appreciated.…”

“…In the applied exponential asymptotics literature, these steps are often associated with ensuring that the ansatz (4.9) is 'consistent with the low-order perturbative expansion'. 14 Let us suppose that w = χ(z) lies on the same branch as our perturbative germ at w = 0. From the general arguments of section 3.2 we know that χ(z) must be zero at singularities of the low-order terms y 1 (z), y 2 (z), .…”

Capturing the cascade: a transseries approach to delayed bifurcations

“…Transseries and summation techniques have been used to study the behaviour of a wide range of parameter-dependent continuous systems. Applications include the study of general nonlinear ordinary differential equations [12,13,42,48], the first Painlevé equation [4,35,49], topological string theory [18,38], field theory and semi-classical quantum mechanics [3,8,23,26,37,44], relativistic hydrodynamics and Einstein partial differential equations [2,11,41], and q−series and knot invariants [25,34]. More recently transseries methods have been been extended to study of discrete problems, such as particular matrix models governed by the first discrete Painlevé equation [4,19,45,50,51].…”

“…It has been shown in [42] that transseries approaches may be used to improve upon asymptotic results obtained using matched asymptotic expansions. In that study, transseries resummation methods were used to obtain a uniform approximation to a continuous problem that had been previously solved using multiple scales methods.…”