A PANORAMIC VIEW OF ASYMPTOTICS R. WONG CITY UNIVERSITY OF HONG KONG FoCM 2008.
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Transcript of A PANORAMIC VIEW OF ASYMPTOTICS R. WONG CITY UNIVERSITY OF HONG KONG FoCM 2008.
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.
3
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,
4
1. INTEGRAL METHODS
2. DIFFERENTIAL EQUATION THEORY
3. EXPONENTIAL ASYMPTOTICS
4. SINGULAR PERTURBATION TECHNIQUES
5. DIFFERENCE EQUATIONS
6. RIEMANN-HILBERT METHOD
5
Steepest descent method (Debye)
I. INTEGRAL METHODS
6
Coalescing Saddle points (Chester, Friedman & Ursell; 1957)
7
Cubic transformation
8
APPLICATIONS
9
F. Ursell, On Kelvin's ship-wave pattern, J. Fluid Mech., 1960
10
11
12
M. V. Berry, Tsunami asymptotics, New J. of Physics, 2005
13
II. DIFFERENTIAL EQUATION THEORY
Liouville transformation:
14
Liouville-Green (WKB) approximation
Double asymptotic feature (Olver, 1960's)
Control function
15
Total variation
16
Rosenlicht, Hardy fields, J. Math. Anal. Appl., 1983.
17
B. Turning point
Langer transformation:
18
Two linearly independent solutions
19
C. Simple pole
Transformation:
20
Bessel-type expansion
21
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.
22
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
23
Stokes’ phenomenon:
Berry’s transition (1989):
24
25
26
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”.
27
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.
28
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.
29
“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”.
30
Boundary – Value Problem
31
32
Example
WKB method gives
Matching technique gives
33
Dendritic Solidification (J. S. Langer, Phys Rev. A, 1986)
(1)
Needle-crystal solution:
(3)
(2)
(4)
34
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
35
Carrier and Pearson I : ODE, 1968
An approximate solution
36
Four approximate solutions
37
Spurious solution :
38
For to be an approximate solution, to leading order we must have
spikes
39
C. G. Lange (1983)
40
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()?
41
V. DIFFERENCE EQUATIONS
42
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.
43
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”.
44
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
45
IV RIEMANN-HILBERT METHOD
46
THEOREM (Fokas, Its and Kitaev, 1992)
47
14
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
48
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.
49
Deift & Zhou’s method of steepest descent
50
FREUD WEIGHTS