Introductory Lectures: Resurgence in Differential Equations, and Effective Methods

Gerald Dunne

University of Connecticut

Isaac Newton Institute Spring School: Asymptotic Methods and Applications, March 22-26, 2021

Isaac Newton Institute Programme: Applicable Resurgent Asymptotics, 2021/2022 Basic Introduction to Resurgence: A Beginner’s Guide

Lecture 3 Basic Introduction to Resurgence: A Beginner’s Guide

1. Lecture 1: Resurgence & Linear Differential Equations I Trans- and Stokes Phenomenon in ODEs I Borel Summation basics I Recovering Non-perturbative Connection Formulas 2. Lecture 2: Resurgence & Nonlinear Differential Equations I Nonlinear Stokes Phenomenon I Painlevé Equation examples I Parametric Resurgence & Phase Transitions 3. Lecture 3: Effective Summation Methods I Probing the Borel Plane Numerically I The Physics of Padé Approximation I Optimal Summation and Effective Summation Methods: A Basic Introduction

• computing perturbative coefficients is difficult • how do we make practical use of resurgence in a really difficult problem where we can only compute a small number of perturbative coefficients ? • consider a series which appears to be asymptotic, with generic leading large order behavior of the coefficients

Γ(a n + b) c ∼ S , n → ∞ n An

• A → location of the leading Borel singularity • b → nature of the leading Borel singularity • a → appropriate expansion variable • the Stokes constant S → normalization Effective Summation Methods: A Basic Introduction

PN cn • given just N terms of such a series n xn+1 (i) how can we sum accurately ? (ii) how can we extrapolate accurately to x = 0 ? (iii) how can we extrapolate into the complex x plane to probe Stokes transitions ? (iv) how can we extract non-perturbative exponentially small effects (connection) ? • resurgence suggests that local analysis of encodes global information • How much global information can be decoded from a FINITE number of perturbative coefficients ? • resurgent functions have orderly structure in Borel plane ⇒ develop extrapolation and summation methods that take advantage of this Effective Summation Methods: A Basic Introduction

• conclusion: it can make a BIG difference how we sum • processing the same perturbative input data in different ways can lead to vastly different levels of precision • the basic toolkit: ratio tests, series acceleration methods (e.g. Richardson), Padé approximants, orthogonal polynomials, Szegö asymptotics, continued fractions, conformal maps, uniformization maps, ... • the good news: many of these are actually very easy to implement → a simple set of exploratory procedures • the quality of the extrapolation of an asymptotic series is governed by the quality of the analytic continuation of the Borel transform Z ∞ f(x) = dt e−x tB[f](t) 0 • lesson 1: it is better to work in the Borel plane Resurgence in Nonlinear Differential Equations

• empirical observation: for “natural problems” the Borel plane often has structure Padé-Borel

• simple & powerful summation/extrapolation method

2N Z ∞ 2N X cn X cn = dt e−xt tn xn+1 n! n=0 0 n=0 Z ∞ −xt = dt e B2N [f](t) 0

• recall that the singularities of B2N [f](t) determine the non-perturbative physics

• but B2N [f](t) is a polynomial !

• as N → ∞, B2N [f](t) develops singularities • Padé is an excellent “low resolution” detector of singularity structures: “Padé-Borel” method Basics of Padé Approximation

• simple and efficient method to analytically continue a series beyond its P2N n • rational approximation to F2N (t) = n cn t

2N RL(t) X P {F } (t) = = c tn + O t2N+1 [L,M] 2N S (t) n M n • completely algorithmic and algebraic (“built-in”)

• near-diagonal Padé: polynomials RN (t) & SN (t) satisfy the same 3-term recursion relation • hence a deep connection to orthogonal polynomials, and their asymptotics (Szegö ...) Basics of Padé Approximation

1. at very high orders Padé can be numerically unstable: a ratio of polynomials with very large coefficients. It is often more stable to convert to an ‘equivalent’ partial fraction

2N PN n N X n n an t X rn cnt ↔ ↔ PN n t − t n n dn t n n

2. truncated series → continued fraction, which often converges in all of C, minus a number of poles/cuts

2N 2N+1 1 2N+1 1+c1 t+···+c2N t +O t = +O t h1 t 1 + h2 t 1+ h t 1+ 3 .. 1 .+h2N t

3. near-diagonal Padé are related to continued fractions Basics of Padé Approximation

• a Padé approximant only has pole singularities. But for practical applications we are often interested in branch point singularities, e.g. Borel plane, critical exponents, ... • Padé represents a branch point as the accumulation point of an arc of poles: cf. potential theory (electrostatics) Im[t] Im[t]

10 10

5 5

Re[t] Re[t] -10 -5 5 10 -10 -5 5 10

-5 -5

-10 -10

1 1 1 1 Figure: Padé poles of 1 , 2πi 1 , and of 1 , 2πi 1 . (1−t) 5 (e 5 −t) 7 (1−t2) 5 (e 5 −t2) 7 Basics of Padé Approximation • symmetry of singularities is important for Padé Im[t] 2

1

Re[t] 5 10 15 -1

-2

Figure: Padé poles of 1/(1 − t + t2)1/5 • but the function has no singularities on the real t axis ! • Stahl (1997): in the limit N → ∞ Padé produces the “minimal capacitor” Im[t] 1.0

0.5

0.0 Re[t] 0.2 0.4 0.6 0.8 1.0

-0.5

-1.0

Figure: 1/t of Padé poles of 1/(1 − t + t2)1/5 Basics of Padé Approximation

1 • amazing sensitivity to exponent 2 1 1 inverse Padé poles of & 1 1 −10−6 (1 − t + t2) 2 (1 − t + t2) 2

Im[t] 1.0

0.5

0.0 Re[t] 0.2 0.4 0.6 0.8 1.0

-0.5

-1.0 • this sensitivity can be used to our advantage ... see later Basics of Padé Approximation

Exercise 3.1: Explore the Padé pole structures for various functions with interesting singularities. Basics of Padé Approximation

• Padé approximants are not very accurate near a branch point and branch cut, because of accumulating poles • Padé obscures repeated repeated singularities along the same line; but this is exactly what happens for resurgent Borel transforms in nonlinear problems • conformal mapping, before Padé, helps resolve these problems • make a into the unit disk, based on the leading Borel singularity/ies Conformal Mapping

• conformal map of cut domains: t → z Im[t] Im[t]

Re[t] Re[t]

Im[t] Im[z]

Re[t] Re[z] Padé-Conformal-Borel Method

• conformally map the Borel plane into the unit disk (based on the leading singularity), and then re-expand and do Padé inside the unit disk ⇒ significant improvement for resurgent Borel transforms • precision can be quantified (Szegö asymptotics) • much better near singularities: higher precision determination of Stokes constants and resolution of the location and nature of the singularity • better extrapolation from x = +∞ to x = 0 Basics of Padé Approximation

1 5  • cut of the Airy Borel function 2F1 6 , 6 , 1, −t

• after the conformal map: Padé-Conformal-Borel: 10-term approximation to (1 + p)−1/3 Padé-Conformal-Borel: 10-term approximation to (1 + p)−1/3

• precision of the Borel transform along the cut Re[B(p)] 4

3

2

1

p -10 -8 -6 -4 -2 -1

-2

-3 blue: exact and Padé-Conformal-Borel approx. (10 terms) red: Padé-Borel approx. (10 terms) black-dashed: Taylor-Conformal-Borel approx. (10 terms) • Padé-Conformal-Borel is generically much more accurate near the singularity and along the cut Padé-Conformal-Borel: 10-term approx. in physical x plane

• this translates into much better extrapolation of the Borel transform in the original physical x variable, from an expansion near x = +∞ down to x ≈ 0 error 10

0.01

10-5

x 0.1 0.2 0.3 0.4 0.5 purple: 10 term x-Padé approx. (log of fractional error) red: 10 term Padé-Borel approx. blue: 10 term Padé-Conformal-Borel approx. black dotted line: 1% fractional error Padé-Conformal-Borel

• the improved precision can be quantified

−(Nx )1/2 cn ∼ Γ(n + α) ⇒ frac. error ∼ e x-Padé 2 1/3 frac. error ∼ e−(N x) Padé-Borel 2 1/3 frac. error ∼ e−(N x) Taylor-Conformal-Borel 4 1/5 frac. error ∼ e−(N x) Padé-Conformal-Borel

• for a chosen precision (e.g., 1% accuracy), with exactly the same input data we can extrapolate from an N-term expansion + at x = +∞ down to xmin → 0 , scaling with N as:

extrapolation xmin scaling truncated series xmin ∼ N −1 x Padé xmin ∼ N −2 Padé-Borel xmin ∼ N −2 Taylor-Conformal-Borel xmin ∼ N −4 Padé-Conformal-Borel xmin ∼ N Padé-Conformal-Borel

• another advantage of the conformal mapping is that it resolves repeated resurgent Borel singularities • recall that in a nonlinear problem a Borel singularity is expected to be repeated in (certain) integer multiples • these can be obscured by Padé’s attempt to represent a cut as an accumulation of poles Conformal Mapping of Borel Plane

• map the doubly-cut Borel p plane to the unit disc p 2 z z = ←→ p = 1 + p1 + p2 1 − z2 Borel plane singularities: Padé-Borel transform

Im[p] 10

5

Re[p] -0.4 -0.2 0.2 0.4

-5

-10

• before conformal map: resurgence is hidden After Conformal Map: Resurgent Poles in z Plane

Im[z] 1.5

1.0

0.5

Re[z] -1.5 -1.0 -0.5 0.5 1.0 1.5

-0.5

-1.0

-1.5

• Conformal map reveals resurgent structure in Borel plane: Borel singularities separated into their resurgent patterns Optimal Method: Uniformizing Map

• an even better procedure: replace the conformal map with a uniformizing map • Optimality Theorem (O. Costin & GD, 2020): given information about the Riemann surface of the Borel transform (known in many cases for resurgent functions), the optimal extrapolation procedure is to use a uniformizing map. • super-precise exploratory tool • singularity elimination: exponential enhancement in the vicinity of the singularities • application: sensitive determination of the location and nature of a singularity, and its ‘Stokes constant’ • permits extrapolation onto higher Riemann sheets Example: summation & extrapolation of Painlevé I

• Painlevé I equation (double-scaling limit of 2d quantum gravity)

y00(x) = 6 y2(x) − x

• large x expansion:   r ∞ !2n x X 30 y(x) ∼ − 1 + c , x → +∞ 6  n 5/4  n=1 (24 x)

• perturbative input data: {c1, c2, . . . , cN } 4 392 6 272 141 196 832 9 039 055 872 { , − , , − , , . . . , a } 25 625 625 390 625 390 625 N • this expansion defines the tritronquée solution to PI Example: summation & extrapolation of Painlevé I

Exercise 3.2: Painlevé I equation: y00(x) = 6 y2(x) − x

(24x)5/4 1. Show that the Écalle critical variable is 30 2. With the ansatz   r ∞ !2n x X 30 y(x) ∼ − 1 + c , x → +∞ 6  n 5/4  n=1 (24 x)

show that the coefficients cn satisfy the recursion formula: n−2 1 X c = −4(n − 1)2c − c c , n ≥ 3 n n−1 2 m n−m m=2 4 392 with c1 = 25 and c2 = − 625 . 3. Show that the large order growth of the coefficients is r   1 9 ! 1 6 n+1 1 8 128 cn ∼ (−1) Γ 2n − 1 − 3  + 3  5  + ... π 5π 2 2n − 2 2n − 2 2n − 2 Perturbative Large x Expansion is an Asymptotic Series

y(x) 0.5 �=�� �=�� �=�� �=�� �=��

x 5 10 15 20

-0.5

-1.0

-1.5

-2.0

• typical asymptotic series: larger N gets worse at small x Reconstruct global behavior from limited x → +∞ data?

• Painlevé I equation has inherent five-fold symmetry Im[x]

������ ����-������

����-������ Re[x]

����-������ ������

• do our input coefficients (from x = +∞) “know” this ? • most interesting/difficult directions: Stokes transitions Example: summation & extrapolation of Painlevé I

Exercise 3.3: Painlevé I equation: y00(x) = 6 y2(x) − x 1. Show that the Painlevé I equation is invariant under the rescaling x = ζ z and y = ζ−2w where ζ5 = 1. Padé-Borel Transform

• truncate Borel transform expansion at n = N:

N X cn B (p) ≡ p2n−1 N (2n − 1)! n=1

• analytic continuation of BN (p) in complex p plane

PN−1(p) Padé-Borel transform: PBN (p) = QN (p)

• PN−1(p) & QN (p): polynomials • algorithmic: built-in function in Mathematica & Maple • zeros of P and Q encode information about B(p) Im[p] 10

5

Re[p] -0.4 -0.2 0.2 0.4

-5

-10

Figure: Padé poles: N = 50; suggests branch points at p = ±i. Padé-Borel Extrapolation

y(x) x 0.5 1.0 1.5 2.0 2.5 3.0 3.5

-0.2

-0.4

-0.6

-0.8

Figure:( N = 10) black: original expansion; blue-dashed: Padé-Borel extrapolation; black-dotted: analytic result at origin. Increased Precision from Padé-Conformal-Borel Transform

y(x)

-0.18

-0.20

-0.22

-0.24

-0.26

x 0.00 0.05 0.10 0.15 0.20 0.25 0.30

Figure: (N = 10): solid blue curve: extrapolation from Padé-Conformal-Borel transform; dashed blue curve: extrapolation from Padé-Borel transform. Black dotted: solution at x = 0. Padé-Borel transform breaks down on the cut

• most difficult region of Borel p plane: on the cuts

PB50(p)

0.3

0.2

0.1

Im[p] 0.5 1.0 1.5 2.0 2.5 3.0 3.5

-0.1

-0.2

-0.3

• Padé-Borel: no sign of resurgent singularities Conformal-Padé-Borel encodes resurgent behavior on the cut

• most difficult region of Borel p plane: on the cuts

PCB50(p) 0.5

0.4

0.3

0.2

0.1

Im[p] 0.5 1.0 1.5 2.0 2.5 3.0 3.5

-0.1

• Padé-Conformal-Borel: resolves resurgent singularities Conformal-Padé-Borel encodes resurgent behavior on the cut

• zoom-in on 2nd singularity: resurgence implies: !2 1 1 1 r3 jump at second singularity = S2 = ≈ 0.0304... 2 2 π 5

Im[PCB50(p)] Im[p] 1.5 2.0 2.5 3.0 -0.08

-0.09

-0.10

-0.11

-0.12

-0.13

-0.14

-0.15 High Precision at the Origin

• resurgence & Padé-Conformal-Borel transform • “weak coupling to strong coupling” extrapolation • N = 50 terms and Padé-Conformal-Borel input:

y(0) ≈ −0.18755430834049489383868175759583299323116090976213899693337265167... y0(0) ≈ −0.30490556026122885653410412498848967640319991342112833650059344290... y00(0) ≈ 0.21105971146248859499298968451861337073253247206264082468899143841...

 00 2  −65 y (x) − 6y (x) + x x=0 = O(10 )

• Resurgent extrapolation methods encode global information about the function throughout the entire , not just along the positive real axis Nonlinear Stokes Transition: the Tritronquée Pole Region

• Boutroux (1913): asymptotically, the general Painlevé I solution has poles with 5-fold symmetry • Dubrovin conjecture (2009): this asymptotic solution to 2π Painlevé I only has poles in a 5 wedge Im[x]

������ ����-������

����-������ Re[x]

����-������ ������

• proof: O. Costin-Huang-Tanveer (2012) Stokes Transition: Mapping the Tritronquée Pole Region

4π • non-linear Stokes transitions crossing arg(x) = ± 5

Im[x]

5

Re[x] -10 -5 5

-5

Figure: Complex poles: extrapolated (red), trans-asymptotic expression (black). Stokes Transition: Mapping the Tritronquée Pole Region

4π • non-linear Stokes transitions crossing arg(x) = ± 5 • uniformizing map (red) versus conformal map (black) Im[x] 10

5

Re[x] -15 -10 -5 5

-5

-10

• exactly the same input data, just processed differently Fine Structure of the Tritronquée Poles

• Boutroux (1913): general PI solution y(x) is meromorphic throughout complex plane, and poles asymptotically related to those of Weierstrass P:

1 xpole 2 1 3 y(x) ≈ 2 + (x − xpole) + (x − xpole) (x − xpole) 10 6 x2 +h (x − x )4 + pole (x − x )6 + ... pole pole 300 pole • completely different from original asymptotic expansion!

• solution completely determined by xpole and hpole • pole closest to origin for the tritronquée solution

x1 = −2.3841687695688166392991458524493...

h1 = −0.062135739226177640896490141640... Metamorphosis: Asymptotic Series to Meromorphic Function

1 xpole 2 1 3 y(x) = 2 + (x − xpole) + (x − xpole) (x − xpole) 10 6 x2 +h (x − x )4 + pole (x − x )6 + ... pole pole 300 pole

• the extrapolation (yN (x) with N = 50) near 1st pole: 0.9999999999999999999999999999999999997886 y(x) ≈ 2 (x − x1) −35 −34 +3.5 × 10 − 2.4 × 10 (x − x1) 2 −0.238416876956881663929914585244923803(x − x1) 3 +0.166666666666666666666666666666657864(x − x1) 4 −0.06213573922617764089649014164005140(x − x1) −31 5 +4 × 10 (x − x1) 6 +0.0189475357392909503157755851627665(x − x1) + ...

• estimate approx 30 digit precision for x1 and h1 Effective Summation and Extrapolation

overall conclusion: there are relatively simple procedures for improving the summation of a given number of terms of an asymptotic series, by obtaining precise numerical information about the analytic structure of the Borel transform A Few Selected References

I Digital Library of Mathematical Functions: https://dlmf.nist.gov/ I C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I O. Costin, Asymptotics and Borel Summability I M. Mariño, Instantons and Large N : An Introduction to Non-Perturbative Methods in Quantum Field Theory I M. V. Berry and C. J. Howls, “Hyperasymptotics for integrals with saddles”, Proc. R. Soc. A 434, 657 (1991) I G. Edgar, “Trans-series for beginners”, arXiv:0801.4877 I P. A. Clarkson, “Painlevé Equations: Nonlinear ” I D. Dorigoni, “An Introduction to Resurgence, Trans-Series and Alien Calculus,” arXiv:1411.3585. I I. Aniceto, G. Basar and R. Schiappa, “A Primer on Resurgent Transseries and Their Asymptotics,” arXiv:1802.10441. I O.Costin, G. V. Dunne, “Physical Resurgent Extrapolation”, arXiv:2003.07451, “Resurgent Extrapolation: Rebuilding a Function from Asymptotic Data. Painleve I”, arXiv:1904.11593