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A PANORAMIC VIEW OF ASYMPTOTICS R. WONG CITY UNIVERSITY OF HONG KONG FoCM 2008

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2 Gian-Carlo Rota: “Indiscrete Thoughts”, 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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3 Abel (1829): “Divergent series are the invention of the devil” Acta Math 8 (1886), pp Poincaŕe, Sur les integrals irregulières des equations linéaires,

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4 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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5 Steepest descent method (Debye) I. INTEGRAL METHODS

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

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7 Cubic transformation

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8 APPLICATIONS

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

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

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13 II. DIFFERENTIAL EQUATION THEORY Liouville transformation:

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14 Liouville-Green (WKB) approximation Double asymptotic feature (Olver, 1960's) Control function

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15 Total variation

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

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17 B. Turning point Langer transformation:

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18 Two linearly independent solutions

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19 C. Simple pole Transformation:

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20 Bessel-type expansion

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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.

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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

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23 Stokes’ phenomenon: Berry’s transition (1989):

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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”.

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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.

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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”. 3 rd 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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29 “This success is probably most surprising to rigor-oriented mathematicians (or applied mathematicians) when they realize that there still exists no theorem which speaks to the validity or the accuracy 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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30 Boundary – Value Problem

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32 Example WKB method gives Matching technique gives

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33 Dendritic Solidification (J. S. Langer, Phys Rev. A, 1986) (1) Needle-crystal solution: (3) (2) (4)

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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), 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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35 Carrier and Pearson I : ODE, 1968 An approximate solution

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36 Four approximate solutions

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37 Spurious solution :

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

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39 C. G. Lange (1983)

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40 Question 1. Does there exist a unique solution u i (x, ) which is uniformly approximated by ũ i (x, ) in the whole interval [-1, 1]? Question 2. In what sense does ũ i (x, ) approximate u i (x, )? For instance, it is true that for all x [-1, 1]? |u i (x, ) - ũ i (x, )| K Question 3. If n( ) denotes the number of internal spikes, is there a rough estimate for n( )?

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41 V. DIFFERENCE EQUATIONS

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42 J. Wimp, Book Review, Mathematics of Computation, Vol. 56, January issue, 1991, 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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43 1.C. M. Adams, On the irregular cases of linear ordinary difference equations, Trans. A.M.S., 30 (1928), pp G. D. Birkhoff, General Theory of linear difference equations, Trans. A.M.S., 12 (1911), pp G. D. Birkhoff, Formal theory of irregular linear difference equations, Acta Math., 54 (1930), pp G. D. Birkhoff and W. J. Trjitzinsky, Analytic theory of singular difference equations, Acta Math., 60 (1932), pp 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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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

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45 IV RIEMANN-HILBERT METHOD

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46 THEOREM (Fokas, Its and Kitaev, 1992)

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Deift and Zhou, Steepest Descent Method for Riemann-Hilbert Problem,, Ann. Math., 1993, Deift et al, Strong Asymptotics of Orthogonal Polynomials with Respect to Exponential Weights, Comm. Pure and Appl. Math, 1999, Deift et al, Uniform Asymptotics for Polynomials Orthogonal with Respect to Varying Exponential Weights, and, Comm. Pure and Appl. Math., 1999, DEIFT-ZHOU’s METHOD … …… Plancheral-Rotach-type asymptotics

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48 4.Bleher and Its, Semiclassical Asymptotics of Orthogonal Polynomials, Riemann-Hilbert Problem, and Ann. Math., 1999, …, 5.Kriecherbauer and McLaughlin Strong Asymptotics of Polynomials Orthogonal with Respect to Freud Weights, IMRN, 1999,

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49 Deift & Zhou’s method of steepest descent

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50 FREUD WEIGHTS

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