A PANORAMIC VIEW OF ASYMPTOTICS

R. WONG

CITY UNIVERSITY OF HONG KONG

FoCM 2008

2

Gian-Carlo Rota:

“Indiscrete Thoughts”, 1996.

p. 222

One remarkable fact of applied mathematics is the

ubíquitious appearance of divergent series, hypocritically

renamed asymptotic expansions. Isn’t it a scandal that we

teach convergent series to our sophomores and do not tell

them that few, if any, of the series they will meet will

converge? The challenge of explaining what an

asymptotic expansion is ranks among the outstanding but

taboo problems of mathematics.

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Abel (1829):

“Divergent series are the invention of the devil”

Acta Math 8 (1886), pp. 295-344.

Poincaŕe, Sur les integrals irregulières des equations linéaires,

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1. INTEGRAL METHODS

2. DIFFERENTIAL EQUATION THEORY

3. EXPONENTIAL ASYMPTOTICS

4. SINGULAR PERTURBATION TECHNIQUES

5. DIFFERENCE EQUATIONS

6. RIEMANN-HILBERT METHOD

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Steepest descent method (Debye)

I. INTEGRAL METHODS

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Coalescing Saddle points (Chester, Friedman & Ursell; 1957)

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Cubic transformation

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APPLICATIONS

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F. Ursell, On Kelvin's ship-wave pattern, J. Fluid Mech., 1960

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11

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M. V. Berry, Tsunami asymptotics, New J. of Physics, 2005

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II. DIFFERENTIAL EQUATION THEORY

Liouville transformation:

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Liouville-Green (WKB) approximation

Double asymptotic feature (Olver, 1960's)

Control function

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Total variation

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Rosenlicht, Hardy fields, J. Math. Anal. Appl., 1983.

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B. Turning point

Langer transformation:

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Two linearly independent solutions

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C. Simple pole

Transformation:

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Bessel-type expansion

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Langer (1935): in a shrinking neighborhood

pole.

Dunster (1994): coalescence of a turning point and a simple

Olver (1975): Coalescing turning points

Swanson (1956) and Olver (1956, 1958): in fixed intervals.

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III. EXPONENTIAL ASYMPTOTICS

a. Kruskal and Segur (1989), Asymptotics beyond all orders in a model of crystal growth, Stud. Appl. Math.

b. Berry (1989), Uniform asymptotic smoothing of Stokes’ discontinuities, Proc. Roy. Soc. Lond. A

Airy function

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Stokes’ phenomenon:

Berry’s transition (1989):

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Stokes (1857),

On the discontinuity of arbitrary constants which

appear in divergent developments.

Stokes (1902) : Survey paper

“The inferior term enters as it were into a mist, is

hidden for a little from view, and comes out with

its coefficients changed”.

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Resurgence

Optimal truncation

Berry & Howls (1990): hyperasymptotics and superasymptotics.

Hyperasymptotics – re-expanding the remainder terms in

optimally truncated asymptotic series.

Superasymptotics – exponentially improved asymptotic

expansion.

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VI. SINGULAR PERTURBATION TECHNIQUES

Sydney Goldstein, Fluid Mechanics in the first half of this

century, Annual Review Fluid Mechanics, 1 (1969), 1 – 28 ;

“The paper will certainly prove to be one of the most extraordinary

papers of this century, and probably of many centuries”.

3rd International Congress of Mathematicians, Heidelberg (1904) ,

Ludwig Prandtl, On fluid motion with small friction ,

ICM (1905), Vol. 3, pp. 484 – 491.

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“This success is probably most surprising to rigor-oriented math

ematicians (or applied mathematicians) when they realize that th

ere still exists no theorem which speaks to the validity or the acc

uracy of Prandtl’s treatment of his boundary-layer problem; but

seventy years of observational experience leave little doubt of its

validity and its value”.

G. F. Carrier, Heuristic Reasoning in Applied

Mathematics, Quart. Appl. Math., 1972, pp. 11 – 15;

Special Issue : Symposium on “The Future of

Applied Mathematics”.

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Boundary – Value Problem

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Example

WKB method gives

Matching technique gives

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Dendritic Solidification (J. S. Langer, Phys Rev. A, 1986)

(1)

Needle-crystal solution:

(3)

(2)

(4)

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N being some positive constant

That is, there is no needle-crystal solution.

Kruskal and Segur (1989): Asymptotics beyond all orders in a model of crystal growth, Studies in Appl. Math., 85(1991), 129-151.

Amick and McLeod (1989): A singular perturbation problem in needle crystals, Arch. Rat. Mech & Anal.

Langer conjectured: the solution to (1) with boundary conditions in (2) and (3) satisfies

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Carrier and Pearson I : ODE, 1968

An approximate solution

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Four approximate solutions

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Spurious solution :

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For to be an approximate solution, to leading order we must have

spikes

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C. G. Lange (1983)

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Question 1. Does there exist a unique solution ui(x, )

which is uniformly approximated by ũi(x, )

in the whole interval [-1, 1]?

Question 2. In what sense does ũi(x, ) approximate

ui(x, )? For instance, it is true that

for all x [-1, 1]?

|ui(x, ) - ũi(x, )| K

Question 3. If n() denotes the number of internal spikes,

is there a rough estimate for n()?

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V. DIFFERENCE EQUATIONS

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J. Wimp, Book Review, Mathematics of Computation, Vol. 56, January issue, 1991, 388-396.

There are still vital matters to be resolved in asymptotic analysis. At least one widely quoted theory, the asymptotic theory of irregular difference equations expounded by G. D. Birkhoff and W. R. Trjitzinsky [5, 6] in the early 1930’s, is vast in scope; but there is now substantial doubt that the theory is correct in all its particulars. The computations involved in the algebraic theory alone (that is, the theory that purports to show there are a sufficient number of solutions which formally satisfy the difference equation in question) are truly mindboggling.

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1. C. M. Adams, On the irregular cases of linear ordinary difference equations, Trans. A.M.S., 30 (1928), pp. 507-541.

2. G. D. Birkhoff, General Theory of linear difference equations, Trans. A.M.S., 12 (1911), pp. 243-284.

3. G. D. Birkhoff, Formal theory of irregular linear difference equations, Acta Math., 54 (1930), pp. 205-246.

4. G. D. Birkhoff and W. J. Trjitzinsky, Analytic theory of singular difference equations, Acta Math., 60 (1932), pp. 1-89.

Frank Olver: “the work of B & T set back all research into the asymptotic solution of difference equations for most of the 20th Century”.

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Question 1. What is a turning point for a second-order linear

difference equation?

2. How does Airy’s function arise from a 3-term

3. How the function ζin Ai(λ ζ) is obtained,

recurrence relation, when the function itself

does not satisfy any difference equation.

such as Langer’s transformation for differential

equations or cubic transformations for integrals.

when there is no corresponding transformation

2/3

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IV RIEMANN-HILBERT METHOD

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THEOREM (Fokas, Its and Kitaev, 1992)

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1. Deift and Zhou, Steepest Descent Method for Riemann-Hilbert Problem, , Ann. Math., 1993, 295-368.

2. Deift et al, Strong Asymptotics of Orthogonal Polynomials with Respect to Exponential Weights, Comm. Pure and Appl. Math, 1999, 1491-1552.

3. Deift et al, Uniform Asymptotics for Polynomials Orthogonal with Respect to Varying Exponential Weights, and , Comm. Pure and Appl. Math., 1999, 1335-1425.

DEIFT-ZHOU’s METHOD

…

……

Plancheral-Rotach-type asymptotics

Plancheral-Rotach-type asymptotics

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4. Bleher and Its, Semiclassical Asymptotics of Orthogonal Polynomials, Riemann-Hilbert Problem, and Ann. Math., 1999, 185-266.

… ,

5. Kriecherbauer and McLaughlin Strong Asymptotics of Polynomials Orthogonal with Respect to Freu

d Weights, IMRN, 1999, 299-333.

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Deift & Zhou’s method of steepest descent

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FREUD WEIGHTS