and Hyperasymptotic Series—

John P. Boyd †

University of Michigan

Abstract. Singular perturbation methods, such as the method of multiple scales

and the method of matched asymptotic expansions, give series in a small parameter

which are asymptotic but (usually) divergent. In this survey, we use a plethora of

examples to illustrate the cause of the divergence, and explain how this knowledge

can be exploited to generate a ”hyperasymptotic” approximation. This adds a second

asymptotic expansion, with di¬erent scaling assumptions about the size of various

terms in the problem, to achieve a minimum error much smaller than the best

possible with the original asymptotic series. (This rescale-and-add process can be

repeated further.) Weakly nonlocal solitary waves are used as an illustration.

Key words: Perturbation methods, asymptotic, hyperasymptotic, exponential small-

ness

AMS: 34E05, 40G99, 41A60, 65B10

“Divergent series are the invention of the devil, and it is shameful

to base on them any demonstration whatsoever.”

” Niels Hendrik Abel, 1828

1. Introduction

2. The Necessity of Computing Exponentially Small Terms

3. De¬nitions and Heuristics

4. Optimal Truncation and Superasymptotics for the Stieltjes Func-

tion

5. Hyperasymptotics for the Stieltjes Function

6. A Linear Di¬erential Equation

7. Weakly Nonlocal Solitary Waves

8. Overview of Hyperasymptotic Methods

9. Isolation of Exponential Smallness

—

This work was supported by the National Science Foundation through grant

OCE9119459 and by the Department of Energy through KC070101.

†

2 John P. Boyd

10. Darboux™s Principle and Resurgence

11. Steepest Descents

12. Stokes Phenomenon

13. Smoothing Stokes Phenomenon: Asymptotics of the Terminant

14. Matched Asymptotic Expansions in the Complex Plane: The PKKS

Method

15. Snares and Worries: Remote but Dominant Saddle Points, Ghosts,

Interval-Extension and Sensitivity

16. Asymptotics as Hyperasymptotics for Chebyshev, Fourier and Oth-

er Spectral Methods

17. Numerical Methods for Exponential Smallness or: Poltergeist-Hunting

by the Numbers, I: Chebyshev & Fourier Spectral Methods

18. Numerical Methods, II: Sequence Acceleration and Pad´ and Hermite-

e

Pad´ Approximants

e

19. High Order Hyperasymptotics versus Chebyshev and Hermite-Pad´

e

Approximations

20. Hybridizing Asymptotics with Numerics

21. History

22. Books and Review Articles

23. Summary

1. Introduction

Divergent asymptotic series are important in almost all branch-

es of physical science and engineering. Feynman diagrams (particle

physics), Rayleigh-Schroedinger perturbation series (quantum chem-

istry), boundary layer theory and the derivation of soliton equations

(¬‚uid mechanics) and even numerical algorithms like the “Nonlinear

Galerkin” method [66, 196] are examples. Unfortunately, classic texts

like van Dyke [297], Nayfeh [229] and Bender and Orszag [19], which

are very good on the mechanics of divergent series, largely ignore two

important questions. First, why do some series diverge for all non“zero

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3

Exponential Asymptotics

Table I. Non-Soliton Exponentially Small Phenomena

Phenomena Field References

Dendritic Crystal Growth Condensed Matter Kessler, Koplik & Levine [163]

Kruskal&Segur[171, 172]

Byatt-Smith[86]

Viscous Fingering Fluid Dynamics Shraiman [276]

(Sa¬man-Taylor Problem) Combescot et al.[103]

Hong & Langer [146]

Tanveer [288, 289]

Di¬usion& Merger Reaction-Di¬usion Carr [92], Hale [137],

of Fronts Systems Carr & Pego [93]

on an Exponentially Fusco & Hale [130]

Long Time Scale Laforgue& O™Malley

[173, 174, 175, 176]

Superoscillations in Applied Mathematics, Berry [31, 32]

Fourier Integrals, Quantum Mechanics,

Quantum Billiards, Electromagnetic Waves

Gaussian Beams

Rapidly-Forced Classical Chang [94]

Pendulum Physics Scheurle et al. [275]

Resonant Sloshing Fluid Mechanics Byatt-Smith & Davie [88, 89]

in a Tank

Laminar Flow Fluid Mechanics, Berman [23], Robinson [272],

in a Porous Pipe Space Plasmas Terrill [290, 291],

Terrill & Thomas [292],

Grundy & Allen [135]

Je¬rey-Hamel ¬‚ow Fluid Mechanics Bulakh [85]

Stagnation points Boundary Layer

Shocks in Nozzle Fluid Mechanics Adamson & Richey [2]

Slow Viscous Flow Past Fluid Mechanics Proudman & Pearson [264],

Circle, Sphere (Log & Power Series) Chester & Breach [98]

Skinner [283]

Kropinski,Ward&Keller[170]

Log-and-Power Series Fluids, Electrostatic Ward,Henshaw

&Keller[308]

Log-and-power series Elliptic PDE on Lange&Weinitschke[179]

Domains with Small Holes

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4 John P. Boyd

Table I. Non-Soliton Exponentially Small Phenomena (continued)

Phenomena Field References

Equatorial Kelvin Wave Meteorology, Boyd & Christidis [74, 75]

Instability Oceanography Boyd&Natarov[76]

Error: Midpoint Rule Numerical Analysis Hildebrand [143]

Radiation Leakage from a Nonlinear Optics Kath & Kriegsmann [162],

Fiber Optics Waveguide Paris & Wood [258]

Liu&Wood[183]

Particle Channeling Condensed Matter Dumas [119, 120]

in Crystals Physics

Island-Trapped Oceanography Lozano&Meyer [185],

Water Waves Meyer [210]

Chaos Onset: Physics Holmes, Marsden

Hamiltonian Systems & Scheurle [145]

Separation of Separatrices Dynamical Systems Hakim & Mallick [136]

Slow Manifold Meteorology Lorenz & Krishnamurthy [184],

in Geophysical Fluids Oceanography Boyd [65, 66]

Nonlinear Oscillators Physics Hu[149]

ODE Resonances Various Ackerberg&O™Malley[1]

Grasman&Matkowsky[133]

MacGillivray[191]

French ducks (“canards”) Various MacGillivray&Liu

&Kazarino¬[192]

where is the perturbation parameter? And how can one break the

“Error Barrier” when the error of an optimally-truncated series is too

large to be useful?

This review o¬ers answers. The roots of hyperasymptotic theory

go back a century, and the particular example of the Stieltjes function

has been well understood for many decades as described in the books of

Olver [249] and Dingle [118]. Unfortunately, these ideas have percolated

only slowly into the community of derivers and users of asymptotic

series.

I myself am a sinner. I have happily applied the method of multiple

scales for twenty years [67]. Nevertheless, I no more understood the

reason why some series diverge than why my son is lefthanded.

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5

Exponential Asymptotics

Table II. Selected Examples of Exponentially Small Quantum Phenomena

Phenomena References

Energy of a Quantum Fr¨man, [128]

o

ˇ´zek et al.[100]

+

Double Well (H2 , etc.) Ciˇ

Harrell[141, 142, 140]

Imaginary Part of Eigenvalue Oppenheimer [255],

of a Metastable Reinhardt [269] ,

Quantum Species: Hinton & Shaw [144],

Stark E¬ect Benassi et al.[18]

(External Electric Field)

Im(E): Cubic Anharmonicity Alvarez [6]

ˇ iˇ

C´zek & Vrscay [101]

Im(E): Quadratic Zeeman E¬ect

(External Magnetic Field)

Transition Probability, Berry & Lim [42]

Two-State Quantum System

(Exponentially Small in

Speed of Variations)

Width of Stability Bands Weinstein & Keller

for Hill™s Equation [313, 314]

Above-the-Barrier Pokrovskii

Scattering & Khalatnikov [262]

Hu&Kruskal[152, 150, 151]

Anosov-perturbed cat map: semiclassical asymptotics Boasman&Keating[46]

In this review, we shall concentrate on teaching by examples. To

make the arguments accessible to a wide readership, we shall omit

proofs. Instead, we will discuss the key ideas using the same tools of