http://dx.doi.org/10.1090/gsm/075

Applied Asymptotic Analysis

Peter D. Miller

Graduate Studies in Mathematics Volum e 75

^^% lP2p\ | America n Mathematica l Societ y Providence , Rhod e Islan d This page intentionally left blank Applied Asymptotic Analysis

Peter D. Miller

Graduate Studies in Mathematics Volum e 75

^^% lP2p\ | America n Mathematica l Societ y Providence , Rhod e Islan d Editorial Board Walter Craig Nikolai Ivanov Steven G. Krantz David Saltman (Chair)

2000 Mathematics Subject Classification. Primary 34E05, 34E10, 34E13, 34E15, 34E20, 41A60, 74J30, 33-01, 81Q20.

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Library of Congress Cataloging-in-Publication Data Miller, Peter D. (Peter David), 1967- Applied asymptotic analysis / Peter D. Miller. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 75) Includes bibliographical references and index ISBN 0-8218-4078-9 (alk. paper) 1. Asymptotic expansions. 2. Differential equations—Asymptotic theory. 3. Approximation theory. 4. Integral equations—Asymptotic theory. I. Title. II. Series. QA431.M477 2006 511'.4—dc22 2006040794

Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2006 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 11 10 09 08 07 06 To my teachers This page intentionally left blank Contents

Preface

Part 1. Fundamentals Chapter 0. Themes of Asymptotic Analysis §0.1. Theme: Asymptotics, Convergent and Divergent Asymptotic Series §0.2. Theme: Other Parameters and Nonuniformity 0.2.1. First example. Oscillations 0.2.2. Second example. Boundary layers §0.3. Theme: Differential Equations §0.4. Theme: Universal Partial Differential Equations and Canonical Physical Models Chapter 1. The Nature of Asymptotic Approximations §1.1. Asymptotic Approximations and Errors 1.1.1. Order relations among functions 1.1.2. Statements following from the order relations 1.1.3. Absolute and relative errors §1.2. Convergent versus Asymptotic Series: Concepts 1.2.1. Convergent power series 1.2.2. Introduction to asymptotic series §1.3. Asymptotic Sequences and Series: General Definitions §1.4. How to "Sum" an Asymptotic Series §1.5. Asymptotic Root Finding 1.5.1. A regular perturbation problem vm Contents

1.5.2. A singular perturbation problem. Rescaling and the principle of dominant balance 40 §1.6. Notes and References 43

Part 2. Asymptotic Analysis of Exponential Integrals

Chapter 2. Fundamental Techniques for Integrals 47 §2.1. Review of Basic Methods 47 §2.2. Exponential Integrals and Watson's Lemma 52 §2.3. Elementary Generalizations of Watson's Lemma 56

Chapter 3. Laplace's Method for Asymptotic Expansions of Integrals 61 §3.1. Introduction 61 §3.2. Nonlocal Contributions 62 §3.3. Contributions from Endpoints 64 §3.4. Contributions from Interior Maxima 67 §3.5. Summary of Generic Leading-order Behavior 70 §3.6. Application: Weakly Diffusive Regularization of Shock Waves 73 3.6.1. The method of characteristics 75 3.6.2. Regularization of shocks by diffusion. Burgers' equation 78 3.6.3. The Cole-Hopf transformation and the solution of the initial-value problem for Burgers' equation 80 3.6.4. Analysis of the solution in the limit of vanishing diffusion 82 §3.7. Multidimensional Integrals 87 §3.8. Notes and References 93

Chapter 4. The Method of Steepest Descents for Asymptotic Expansions of Integrals 95 §4.1. Introduction 95 §4.2. Contour Deformation 97 §4.3. Paths of Steepest Descent 98 §4.4. Saddle Points 103 §4.5. Parametrization-independent Local Contributions 107 §4.6. Application: Long-time Asymptotic Behavior of Diffusion Processes 108 4.6.1. A derivation of the diffusion equation 109 Contents IX

4.6.2. Solution of the diffusion equation and the corresponding initial-value problem 110 4.6.3. Long-time asymptotics via the method of steepest descents 112 §4.7. Application: Asymptotic Behavior of Special Functions, Airy Functions and the Stokes Phenomenon 116 4.7.1. Integral representations for Airy functions 116 4.7.2. Preliminary transformations necessary for asymptotic analysis of Ai(x) for large x 117 4.7.3. Determination of the path. Dependence of the path on K 119 4.7.4. Asymptotic behavior of Ai(x) for large x. The Stokes phenomenon 122 §4.8. The Effect of Branch Points 125 4.8.1. Application: Asymptotics of transform integrals 135 4.8.2. Application: Selection of particular solutions of linear differential equations admitting integral representations 142 §4.9. Notes and References 147

Chapter 5. The Method of Stationary Phase for Asymptotic Analysis of Oscillatory Integrals 149 §5.1. Introduction 149 §5.2. Nonlocal Contributions 151 §5.3. Contributions from Interior Stationary Phase Points 156 5.3.1. Putting the exponent in normal form by a change of variables 156 5.3.2. Analysis of Ji(A) by the method of steepest descents 158 5.3.3. Analysis of J2(A) using integration by parts 160 5.3.4. The asymptotic contribution of a stationary phase point 161 §5.4. Summary of Generic Leading-order Behavior 162 §5.5. Application: Long-time Behavior of Linear Dispersive Waves 164 5.5.1. Partial differential equations for linear dispersive waves 164 5.5.2. Analysis of the solution formula. Long• time asymptotics using the method of stationary phase 167 5.5.3. Structure of the wave field for large time. Modulated wavetrains and group velocity 169 §5.6. Application: Semiclassical Dynamics of Free Particles in Quantum Mechanics 171 5.6.1. Derivation of the dispersion relation for "matter waves" 171 Contents

5.6.2. The Schrodinger equation for a free particle. Interpretation of the Schrodinger wave function 173 5.6.3. The semiclassical limit. Heuristic reasoning 174 5.6.4. Rigorous semiclassical asymptotics using the method of stationary phase 177 §5.7. Multidimensional Integrals 181 §5.8. Notes and References 193

Part 3. Asymptotic Analysis of Differential Equations Chapter 6. Asymptotic Behavior of Solutions of Linear Second- order Differential Equations in the Complex Plane 197 §6.1. Qualitative Theory of Solutions 198 6.1.1. Reduction to canonical form 198 6.1.2. Solutions viewed as analytic functions of the complex variable z 200 6.1.3. Reduction of order 213 §6.2. Asymptotic Behavior near Ordinary and Regular Singular Points 214 6.2.1. Series solutions at ordinary points 215 6.2.2. Series solutions at regular singular points. The method of Frobenius 216 §6.3. Asymptotic Behavior near Irregular Singular Points 223 6.3.1. Formal asymptotic series 223 6.3.2. Existence of true solutions described by the formal asymptotic series. The Stokes phenomenon 229 6.3.3. Another approach to the existence of true solutions and the Stokes phenomenon. Borel summation 246 §6.4. Notes and References 251 Chapter 7. Introduction to Asymptotics of Solutions of Ordinary Differential Equations with Respect to Parameters 253 §7.1. Regular Perturbation Problems 254 7.1.1. Formal power series expansions 255 7.1.2. Solving for yn(x). Variation of parameters 256 7.1.3. Justification of the formal expansion 260 §7.2. Singular Asymptotics 263 7.2.1. The WKB method 263 7.2.2. The special case of an asymptotic power series for f{x-X) 268 7.2.3. Turning points 277 Contents XI

7.2.4. Problems with more than one turning point. The Bohr- Sommerfeld quantization rule 300 7.2.5. Uniform asymptotics near turning points. Langer transformations 304 §7.3. Notes and References 310 Chapter 8. Asymptotics of Linear Boundary-value Problems 311 §8.1. Asymptotic Existence of Solutions 312 8.1.1. Case I: a(x) ^ 0 on [a, ft] and e is positive but sufficiently small 314 8.1.2. Case II: b(x) - a\x)/2 < 0 on [a,/?] and e is positive 314 §8.2. An Exactly Solvable Boundary-value Problem: Phenomenology of Boundary Layers 315 §8.3. Outer Asymptotics 318 §8.4. Rescaling and Inner Asymptotics for Boundary Layers and Internal Layers 321 §8.5. Matching of Asymptotic Expansions, Intermediate Variables, and Uniformly Valid Asymptotics 325 §8.6. Examples 328 §8.7. Proving the Validity of Uniform Approximations 342 §8.8. The Method of Multiple Scales 350 §8.9. Notes and References 353 Chapter 9. Asymptotics of Oscillatory Phenomena 355 §9.1. Perturbation Theory in Linear Algebra and Eigenvalue Problems 356 9.1.1. Nondegenerate theory 357 9.1.2. Degenerate theory 362 9.1.3. More on solvability conditions. Inner products and adjoint s 365 §9.2. Periodic Boundary Conditions and Mathieu's Equation 368 9.2.1. Floquet theory 368 9.2.2. Periodic and antiperiodic solutions. Formal asymptotics 371 9.2.3. Justification of the expansions 377 §9.3. Weakly Nonlinear Oscillations 382 9.3.1. Periodic solutions near equilibrium 383 9.3.2. A perturbative approach to weak cubic nonlinearity. Secular terms 384 9.3.3. Removal of secular terms. Strained coordinates and the Poincare-Lindstedt method 388 Xll Contents

9.3.4. The method of multiple scales 391 9.3.5. Justification of the expansions 397 §9.4. Notes and References 400 Chapter 10. Weakly Nonlinear Waves 401 §10.1. Derivation of Universal Partial Differential Equations Using the Method of Multiple Scales 401 10.1.1. Modulated wavetrains with dispersion and nonlinear effects. The cubic nonlinear Schrodinger equation 402 10.1.2. Spontaneous excitation of a mean flow 410 10.1.3. Multiple wave resonances 417 10.1.4. Long wave asymptotics. The Boussinesq equation and the Korteweg-de Vries equation 423 §10.2. Waves in Molecular Chains 425 10.2.1. The Fermi-Pasta-Ulam model 426 10.2.2. Derivation of the cubic nonlinear Schrodinger equation 427 10.2.3. Derivation of the Boussinesq and Korteweg- de Vries equations 432 §10.3. Water Waves 433 10.3.1. Derivation of the cubic nonlinear Schrodinger equation 436 10.3.2. Derivation of the Korteweg-de Vries equation 444 §10.4. Notes and References 447 Appendix: Fundamental Inequalities 451 Triangle Inequalities 451 Minkowski Inequalities 452 Holder Inequalities 452 Bibliography 453 Index of Names 455

Subject Index 457 Preface

This text is meant to serve as an introduction to the theory and applications of asymptotic approximations. Such approximations are characterized by the property that they are made more accurate by the tuning of an auxiliary parameter. Frequently (but not always) asymptotic approximations arise as the partial sums of a formal power series in a small parameter e, and almost as frequently the formal series is divergent for all e ^ 0. The fact that the series cannot be summed is in no way in contradiction to the utility of the partial sums as asymptotic approximations made more accurate by reducing the magnitude of e. The distinction between such "asymptotic series" and convergent series was established concretely as recently as the end of the nineteenth century, although asymptotic approximations had been in practical use for a very long time before then. This indicates that the distinction between asymptotic and convergent series is a potential point of confusion, especially for students considering series in applications for the first time. These days, it is not long after students begin studying mathematics, physics, engineering, or another of the quantitative applied sciences when they first encounter in their reading an expression of the form XQ + ex\ + e2X2 + - - • that is presented as a solution of some equation for an unknown x involving a small parameter e. On the one hand, students learn quickly how to find the coefficients xn] one just substitutes the expression into the equation to be solved and gathers together the like powers of e. On the other hand, without further analysis of a different sort it is not at all clear what the meaning of the so-obtained formal series actually is. Is the series convergent, asymptotic, both, or neither? What exactly does the calculation of the first few xn's buy? More often than not students are left to interpret the series on

xni XIV Preface their own, assuming that they have been trained to think about the precise meaning of an infinite series in the first place. It is my hope that this text helps to clear up some of the potential confusion associated with asymptotic series (and asymptotic approximations more generally). I also hope that this text will generate interest in the fascinating subject of asymptotic analysis and its applications. This book was written for students with a background in differential equations, advanced calculus (rigorous limits), complex variables, and ma• trix algebra at the level of undergraduate courses. Some of the applica• tions involve partial differential equations and Fourier and Laplace trans• form methods, but for the most part this material is introduced where it is needed. In fact, several topics are introduced in this text that are not ordinarily viewed as part of a course on this subject but that are easy to introduce to students with the specified background. These topics include the method of characteristics for partial differential equations (Chapter 3), as well as the Contraction Mapping Principle (Chapter 6) and pseudoinverse operators (Chapter 9) from functional analysis. The applications and prob• lems touch on the theory of weakly viscous shock waves, quantum mechanics and the semiclassical limit, long-time behavior of diffusion processes and dis• persive waves, how to count lattice points in the plane using Fourier theory, random matrix theory, orthogonal polynomials, zeros of Taylor polynomials for nonvanishing functions, aspects of special functions (in particular the central role in many problems played by the Airy function Ai(z)), nonlin• ear lattices (coupled pendula and the Fermi-Pasta-Ulam model), and water waves. This text was developed over a period of five years and was used four times to teach a first-year graduate course in asymptotic methods. I would like to thank the students who gave me valuable feedback on my course notes as they developed into this text. I also want to specifically acknowledge many useful conversations I had with Charlie Doering, Jeffrey Rauch, and Alexander Tovbis. I have dedicated this book to my teachers. Among many I would like to single out Rich Haberman and Doug Reinelt for getting me started, Marty Greenlee for explaining how it all starts with the beautiful basics of func• tional analysis, Bill Faris for elucidating how quantum mechanics is the premier application thereof, Al Scott for teaching me how to carry out prac• tical calculations in quantum theory, and for showing me the unexpected complexity of discrete systems (e.g. pendulum chains), Dave Levermore for explaining the right way to think about semiclassical limits, Nick Ercolani for showing me the structure and beauty behind integrable systems, Alan Newell for giving me a "big picture" of nonlinear waves, and Hermann Flaschka for Preface xv teaching me how important it is to keep it all honest. Anyone who knows these people will see their footprints on every page of this book. The financial support of the National Science Foundation under grant numbers DMS-0103909 and DMS-0354373 and of the Alfred P. Sloan Foun• dation under a Sloan Research Fellowship was crucial to the completion of this work. But even more important was the moral support of my wife, Connie. I appreciate their help.

Peter D. Miller Ann Arbor January 2006 This page intentionally left blank Bibliography

[1] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series, volume 149, Cambridge University Press, Cambridge, 1991. [2] W. Balser, Formal Power Series and Linear Systems of Meromorphic Ordinary Dif• ferential Equations, Universitext, Springer, New York, 2000. [3] G. D. Birkhoff, "Singular points of ordinary linear differential equations", Trans. Amer. Math. Soc, 10, 436-470, 1909. [4] M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, Sixth edition, Pergamon Press, Oxford, 1980. [5] L. Brandolini, A. Iosevich, and G. Travaglini, "Planar convex bodies, Fourier trans• form, lattice points, and irregularities of distribution", Trans. Amer. Math. Soc, 355, 3515-3535, 2003. [6] G. F. Carrier, M. Krook, and C. E. Pearson, Functions of a Complex Variable, Theory and Applications, Hod Books, Ithaca, New York, 1983. [7] J. D. Cole, Doctoral thesis, 1949. [8] P. G. Drazin and R. S. Johnson, Solitons: An Introduction, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1993. [9] A. Erdelyi, Asymptotic Expansions, Dover Publications, Inc., New York, 1956. [10] L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons, Springer, Berlin, 1987. [11] E. Fermi, J. Pasta, and S. Ulam, "Studies in Nonlinear Problems, I." Los Alamos report LA1940, 1955. Reproduced in Nonlinear Wave Motion, (Ed. A. C. Newell), American Mathematical Society, Providence, RI, 1974. [12] A. R. Forsyth, Theory of Differential Equations, volume 6. Dover Publications, Inc., New York 1959. [An unaltered reprinting of the original edition published by Cam• bridge University Press, London, 1906.] [13] G. Frobenius, J. filr Math., 76, 1873. [14] C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, "Method for solving the Korteweg-de Vries equation", Phys. Rev. Lett., 19, 1095-1097, 1967.

453 454 Bibliography

E. Hille, Ordinary Differential Equations in the Complex Domain, Dover, Mineola, 1997. v E. Hopf, "The partial differential equation ut + uux — fj,uxx , Comm. Pure Appl. Math., 3, 201-230, 1950. J. K. Hunter and J. B. Keller, "Weakly nonlinear high frequency waves", Comm. Pure Appl. Math., 36, 547-569, 1983. E. L. Ince, Ordinary Differential Equations, Dover Publications, Mineola, New York, 1956. A. R. Its and V. Yu. Novokshenov, The Isomonodromic Deformation Method in the Theory of Painleve Equations, Lecture Notes in Mathematics, volume 1191, Springer, Berlin, 1986. Shan Jin, C. D. Levermore, and D. W. McLaughlin, "The semiclassical limit of the defocusing NLS hierarchy", Comm. Pure Appl. Math., 52, 613-654, 1999. K. Johansson, "On Szego's asymptotic formula for Toeplitz determinants and gener• alizations", Bull. Set. Math. (2), 112, 257-304, 1988. S. Kamvissis, K. T.-R. McLaughlin, and P. D. Miller, Semiclassical Soliton Ensembles for the Focusing Nonlinear Schrodinger Equation, Annals of Mathematics Studies, volume 154, Princeton University Press, Princeton, 2003. J. Kevorkian and J. D. Cole, Multiple Scale and Singular Perturbation Problems, Applied Mathematical Sciences, volume 114, Springer, Berlin, 1996. P.-S. Laplace, Theorie Analytique des Probabilites [Analytic Probability Theory], vol• ume I, third edition, 1820. P. D. Lax and C. D. Levermore, "The small dispersion limit of the Korteweg-de Vries equation, I, III, III", Comm. Pure Appl. Math., 36, 253-290, 571-593, 809-829, 1983. J. M. Manley and H. E. Rowe, Proc. IRE, 44, 904-914, 1956. M. L. Mehta, Random Matrices, Second Edition, Academic Press, San Diego, 1991. L. M. Milne-Thompson, Theoretical Hydrodynamics, Dover Publications, Mineola, New York, 1968. A. C. Newell, Solitons in Mathematics and Physics, CBMS-NSF Regional Conference Series in Applied Mathematics, volume 48, SIAM, Philadelphia, 1985. D. R. Smith, Singular-perturbation Theory: An Introduction with Applications, Cam• bridge University Press, Cambridge, 1985. . I. Stakgold, Green's Functions and Boundary Value Problems, Wiley, New York, 1979. A. Tovbis, S. Venakides, and X. Zhou, "On semiclassical (zero dispersion limit) solu• tions of the focusing nonlinear Schrodinger equation", Comm. Pure Appl. Math., 57, 877-985, 2004. R. Varga, Scientific Computation on Mathematical Problems and Conjectures, CBMS- NSF Regional Conference Series in Applied Mathematics, volume 60, SIAM, Philadel• phia, 1990. F. Verhulst, Nonlinear Differential Equations and Dynamical Systems, Second Edi• tion, Springer Verlag, Berlin, 1996. W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Dover, Mi• neola, 2002. H. F. Weber, "Uber die Integration der partiellen Differential-gleichung: d2u/dx2 + d2u/dy2 + k2u = 0", Math. Ann., 1, 1-36, 1869. G. B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974. N. J. Zabusky and M. D. Kruskal, "Interactions of 'solitons' in a collisionless plasma and the recurrence of initial states", Phys. Rev. Lett., 15, 240-243, 1965. Index of Names

Ablowitz, M. J., 448, 453 Forsyth, A. R., 94, 453 Airy, G. B., 116 Fredholm, E. I., 312 Frobenius, G., 216, 251, 453 Balser, W., 252, 453 Banach, S., 233 Gardner, C. S., 449, 453 Birkhoff, G. D., 252, 453 Germer, L. H., 172 Bloch, F., 400 Gordon, W., 448 Bohr, N. H. D., 174 Greene, J. M., 449, 453 Borel, F. E. J. E., 248 Greenlee, W. M., xiv Born, M., 310, 453 Boussinesq, V. J., 424, 448 Bragg, Sir W. L., 260 Haberman, R., xiv Brandolini, L., 193, 453 Hilbert, D., 365 Brillouin, L., 310 Hille, E., 147, 251, 252, 453 de Broglie, L. V. P. R., 172 Hopf, E. F. F., 81, 94, 454 Bromwich, T. J. A., 136 Hugoniot, P. H., 110 Burgers, J. M., 80 Hunter, J. K., 310, 454

Carrier, G. F., 147, 453 Ince, E. L., 251, 252, 454 Clarkson, P. A., 448, 453 Iosevich, A., 193, 453 Cole, J. D., 81, 94, 353, 400, 453, 454 Its, A. R., 252, 454

Davisson, C. J., 172 Jeffreys, H., 310 Doering, C. R., xiv Jin, Shan, 193, 454 Drazin, P. G., 448, 453 Johansson, K., 94, 454 Johnson, R. S., 448, 453 Einstein, A., 173 Ercolani, N. M., xiv Kamvissis, S., 193, 454 Erdelyi, A., 44, 453 Keller, J. B., 310, 454 Euler, L., 3 Kelvin, Lord, 151 Kevorkian, J., 353, 400, 454 Faddeev, L. D., 448, 453 Klein, O., 448 Faris, W. G., xiv Korteweg, D. J., 425 Fermi, E., 426, 449, 453 Kramers, H. A., 310 Flaschka, H., xiv Krook, M., 147, 453 Floquet, G., 368 Kruskal, M. D., 449, 453, 454

455 456 Index of Names

Langer, R. E., 305 Wentzel, G., 310 Laplace, P.-S., 62, 93, 454 Whitham, G. B., 94, 448, 449, 454 Lax, P. D., 310, 454 Wolf, E., 310, 453 Levermore, C. D., xiv, 193, 310, 454 Lindstedt, A., 388 Zabusky, N. J., 449, 454 Lyapunov, A. M., 395 Zhou, X., 193, 454

Manley, J. M, 448, 454 Mathieu, C.-L., 368 McLaughlin, D. W., 193, 454 McLaughlin, K. T.-R., 193, 454 Mehta, M. L., 94, 454 Miller, P. D., 193, 454 Milne-Thompson, L. M., 354, 454 Miura, R. M., 449, 453

Neumann, C. G., 361 Newell, A. C., xiv, 449, 453, 454 Novokshenov, V. Yu., 252, 454

Painleve, P., 252 Pasta, J. R., 426, 449, 453 Pearson, C. E., 147, 453 Planck, M. K. E. L., 173 Poincare, J.-H., 7, 94, 388

Rankine, W. J. M., 110 Rauch, J. B., xiv Rayleigh, Lord, 310 Reinelt, D., xiv Riccati, Conto J., 224 Ritt, J. F., 44 Rowe, H. E., 448, 454

Schrodinger, E., 172 Scott, A. C., xiv Smith, D. R., 353, 354, 454 Sommerfeld, A. J. W., 304 Stakgold, I., 252, 400, 454 Stokes, G. G., 151 Strutt, J. W., see Rayleigh, Lord

Takhtajan, L. A., 448, 453 Thompson, W., see Kelvin, Lord Tovbis, A., xiv, 193, 454 Travaglini, G., 193, 453

Ulam, S., 426, 449, 453

Varga, R., 147, 454 Venakides, S., 193, 454 Verhulst, F., 193, 454 de Vries, G., 425

Wasow, W., 44, 251, 454 Watson, G. N., 53 Weber, H. F., 147, 454 Subject Index

a-helix protein molecules, 425 as a universal model near turning points, 3-wave interaction equations, 419, 448 402 entire analytic nature of solutions of, 199 integral representations for solutions of, Abel's Theorem, 369 116-117 absolute continuity, 151, 378 perturbation of, 288, 291, 293, 295, absolute convergence, 6, 7, 184 306-308 absolute error, 23, 53 Stokes matrices for, 245 of order e(z), 23 amplitude, 166 absolute integrability, 5, 62, 63, 83 analytic continuation, 200-206 local, 113 dependence on path, 206 action, 174 in Borel summation theory, 248 associated with a periodic orbit, 304 of canonical solutions to neighboring quantization of, 304 Stokes rays, 244 additive identity, 235 additive inverse, 236 of the gamma function, 201 of the geometric series, 201, 203 adjoint operator, 359, 365-368, 378 1 2 how to find in practice, 366 of the square root z / , 206 Airy function Ai(z), xiv, 117, 288, 293 persistence of linear independence under, analysis of via steepest descents, 116-123 207 as a model for the wave function near a persistence of solutions of differential caustic, 181 equations under, 207 bound for, 292 repetition of, 205 derivative of Aif(z) uniqueness of, 203, 204, 206 integral representation of, 289 using analytic existence theory, 207—208 large z expansion of, 289 using Taylor series, 201-203 large z expansion of, 122-123, 288 Analytic Existence Theorem, 200 steepest descent paths for analysis of, analytic function, 24, 37, 38, 44 119-122 Analytic Implicit Function Theorem, 37, Airy function Bi(z), 288 357, 361, 378, 381, 382 bound for, 292 approximation theory, 26 derivative of Bi' (z) associative property integral representation of, 289 of addition, 235 large z expansion of, 289 of scalar multiplication, 236 large z expansion of, 288 asymptotic approximation, 29 Airy's equation, 116, 197, 199, 289-291, asymptotic expansion, 6, 29 293, 297, 304, 306, 308 formula for coefficients of, 31 458 Subject Index

notation for, 29 uniformly valid approximate solutions of asymptotic methods, 7 y \f (x: e), 327 asymptotic power series, 8 yunif v ' /' asymptotic property, 49, 50, 214 validity of uniform approximation for, asymptotic sequence, 28, 324, 336 342-349 determination of in the WKB method, well-posed, 311 265 Case I of, see Case I asymptotic series, 29 Case II of, see Case II asymptotic sum, 32 negativity condition for, see negativity asymptotic summability, 32 condition attractor, 397 stability result for, 346 averages, 373, 375 Boussinesq equation, 411, 423-424, 448 for the Fermi-Pasta-Ulam model, 432-433 ballistic motion, 85, 176 nonlinear coefficient j3 for, 415 Banach space, 233, 235, 237, 400 Bragg condition, 260 bell-shaped function, 74 Bragg resonance, 260 beyond all orders, 30, 56, 63, 89, 318, 371 branch, 127 bifurcation points, 14 principal, 16, 124 big-oh, 15-17, 20, 43, 152 branch cut, 16, 124, 127 of one, 19 as a mechanism for avoiding black holes, 78 multivaluedness, 127 blackbody radiation, 173 as a steepest descent path, 128 Bloch functions, 400 emanating from a singular point, 210, Bloch Theorem, 400 244 blow up (logarithmic), 284 branch points blow up (of a singularity), 68, 92, 156, 188 in the method of steepest descents, Bohr-Sommerfeld quantization rule, 125-135 300-304, 377 breaking time, 78, 79 Borel summation, 44, 246-252 de Broglie relation, 172 application to the Stokes phenomenon, Bromwich path, 136 251 Burgers' equation, 78-80, 85, 175 boundary conditions as a local conservation law, 110 conversion to homogeneous form, solution of, 80-82 343-344 discrepancies in, 334, 346-348 cancellation effects, 96 homogeneous, 313, 344 canonical form (of a linear homogeneous no-slip, 354 second-order equation), 198, 199, 313 nonlocal influence of, 327 canonical model problem, 299 of Dirichlet type, 312, 353 canonical solutions of mixed type, 312 analytic continuation of to neighboring of Neumann type, 312 Stokes rays, 244 periodic, 368-382 associated with a Stokes ray, 243 boundary layer, 9, 13, 311, 317, 321 Case I, 314, 344, 349 viscous, 354 Case II, 314-315, 344, 349 boundary-value problems, 311 Cauchy integral formula, 23, 202, 204 classical solutions of, 312 Cauchy sequence, 233, 237 common terms of inner and outer Cauchy's Theorem, 150 N.M / \ one Cauchy-Kovaleskaya method, 445 expansions ymatchWe), 326 Cauchy-Riemann equations, 98, 103, 104 homogeneous, 312 Cauchy-Schwarz inequality, 55, 57, 345, 452 inner expansion for, 317, 318, 324 caustics, 179 nonlinear, 353 celestial mechanics, 400 outer expansion for, 316, 318-321 chain rule, 75, 350, 389, 392, 395, 407, 436 failure of, 317, 321 change of basis, 399 recursion relation of coefficients, 320 characteristic curve, 75, 77 region of validity of, 321 characteristic equation, 377 two-point, 312 Subject Index 459

equation equivalent to, 362 contour infinite-dimensional analogue of, 362 intrinsic geometry of, 107 characteristic equation (for a differential contour deformation, 97-98 equation), 315 contour integrals, 318 characteristic function, 184 contraction mapping, 233, 239, 240, Fourier transform of, 184 272-273, 285 characteristic lines, 75, 76, 78, 84 Contraction Mapping Principle, xiv, x-intercepts of, 83 232-234, 237, 240, 244, 252, 272, 273 multiplicity of through a given point, 78, convergence factors, 34, 44 84 convergent methods, 7 plausible, 86 convergent property, 49, 50, 214 relation to stationary phase points, 179 corner layer, 340, 342 union of, 76 correspondence principle, 174 characteristic polynomial, 356, 362 cosmologists, 79 infinite-dimensional analogue of, 381, 382 cost multiple root of, 362 for using WKB formulae near turning perturbed, 356 points, 285 reduced, 356 coupled pendula, xiv, 402 classes continuum limit of, 402, 403 of exponential solutions, 279 coupled system derivatives of, 286 governing the mean and complex of oscillatory solutions, 279 amplitude, 414 derivatives of, 286 critical points, 83, 84, 87, 150 classical mechanics, 425 complex, 104, 115 modified version of, 304 curvature, 185 closure cutoff function under addition, 235 infinitely differentiable, 153, 186 under scalar multiplication, 236 piecewise linear, 33 coherent phenomena, 426 polynomial, 157, 191 Cole-Hopf transformation, 80-82, 85, 86, 175 d'Alembert formula, 424 dispersive analogue of, 175 degeneracy column space, 358 breaking of, 364 commutative property, 235 diagonal matrix of eigenvalues, 358 compact set, 207 diagonalizable matrix, 358, 362 compactness, 208 differential geometry, 447 complementary function differential operator, 368 in a partition of unity, 154 diffusion, 79, 80 complementary subspace (to the range), limit of vanishing, 82-87 359 definition of global dynamics via, 83 completeness, 233, 237, 365 diffusion constant, 80, 83, 109, 175 complex amplitude, 407 diffusion equation, 80, 81 differential equation for, 401 derivation of, 109-110 complex conjugate memory of initial data in, 115 notation © for, 405 solution of, 82, 110-112 compressible fluid dynamics, 79 diffusion processes confining potential, 93 long-time behavior of, xiv conjugate-transpose matrix, 357, 359 via steepest descents, 108-116 as an adjoint, 367 dispersion, 171, 411 connection formulae, 293-300, 302 strong, 410 connection problem, 279 dispersion relation, 166, 404, 406, 407, 409 ill-posed, 280, 282 admitting a mean flow, 410 well-posed, 280-282, 299 admitting resonant third harmonics, 406 conservation law, 109 for a system with discrete translational local, 109, 422 symmetry, 429 constants of motion, 383, 448 for matter waves, 171-173 continuum limit, 402, 403 for the Boussinesq equation, 411 460 Subject Index

for the Fermi-Pasta-Ulam model, 429 equivalence class, 32 for water waves, 438 error symmetry of, 404 optimization of estimate for, 296 dispersive waves Euclidean inner product, see inner product, long-time behavior of, xiv, 164-171 Euclidean dissipation Euclidean vector length, 87 absence of, 410 Euler characteristic, 183 distributive law, 20 Euler equation (type of second-order linear first, 236 differential equation), 213 second, 236 Euler equations (of fluid dynamics), 354 Divergence Theorem, 182, 185 Euler integral (defining r(z)), 3, 59, 124, as integration by parts, 192 201 divisor, 188 Euler's constant 7, 5 dominant term, 31, 283 even part, 58 double integral, 59 Taylor series of, 59 drag, 354 exponent function dynamical phenomena, 355 modified, 62 exponential integral eigenfunction Laplace type, 62, 83, 95, 107, 113, 238 elimination of, 378, 382 change of variables to obtain, 71 eigenspace, 362 Watson type, 53, 55, 57, 65, 131 complete, 363 exponential integral Ei(z), 26, 27 preferred directions within, 364 asymptotic expansion of, 27, 47, 51 eigenvalue, 356 exponential integral (general), 52, 53 algebraic multiplicity of, 363 exponential transformation, 223 asymptotic expansion of exponentially small, 58, 63, 89, 91, 95, 317, justification of, 361 334, 415 degenerate, 362 geometric multiplicity of, 363 fast time, 391-394, 396 multiple Fermat's principle of least time, 310 unfolding of, 364 Fermat's Theorem (of differential calculus), simple, 357 67 eigenvalue problem Fermi-Pasta-Ulam model, xiv, 426-427, 449 for the stationary Schrodinger equation, derivation of Boussinesq equation for, 300 432-433 perturbed, 356-368 derivation of Korteweg-de Vries equation degenerate theory of, 362-365, 375 for, 432-433 nondegenerate theory of, 357-361 derivation of nonlinear Schrodinger eigenvector, 356 equation for, 427-432 correction to, 357 dispersion relation for, 429 elimination of, 400 nonlinear coefficient f3 for, 432 expansion procedure for, 357-360 finite differentiability justification of, 361-362 as an obstruction in the WKB method, generalized, 363 269 left, 357 finite sub-covering, 208 of the monodromy matrix, see fixed point, 272, 276, 285 monodromy matrix, left (row) Floquet multipliers, 370 eigenvector of Floquet theory, 368-371, 375 eikonal equation, 310 Floquet's Theorem, 400 electron diffraction in nickel crystals, 172 fluid dynamics, 253, 354 elementary functions, 318 fluid turbulence, 80 energy integral, 383 fluid/air interface, 433, 434 level sets of, 384 flux, 109 equilibria, 383 for Burgers' equation, 110 equilibrium of a diffusion process, 109 of the Fermi-Pasta-Ulam model, 427 four-wave mixing, 422 stability of, 429 Fourier series, 390 Subject Index 461

of nonlinear trigonometric expressions, Hamilton's equations, 427 385-386 Hamilton-Jacobi equation, 310 Fourier spectrum Hankel function, 123 transport of energy within, 417 hard/soft springs, 432 Fourier transform, 112, 184, 193 harmonic function, 104, 105 decay of, 184 harmonic oscillator, 384 inverse, 112 harmonics, 385, 393, 405 nonlinear analogue of, 447 fundamental, 393, 405, 407, 411-413, 417 smoothness of, 113 higher, 405, 417 strip of analyticity of, 115 resonant, 393, 396, 405-407, 412, 413 Fourier transform pair, 135 second, 391, 412, 413 Fourier-Laplace integral, 116 third, 406, 413 frame of reference, 409 heat kernel, 82 Fredholm alternative, 312 Heaviside step function, 56 frequency, 166 Hermitian matrix, 360, 362 of nonlinear oscillations Hessian matrix, 87, 186 amplitude dependence of, 387 negative definite, 88 fundamental period, 375, 376, 387 Hilbert space, 365, 366, 368, 378, 400 fundamental solution matrix, 399 finite-dimensional, 367 Fundamental Theorem of Algebra, 41, 106 Hilbert's 7th problem, 6 Fundamental Theorem of Calculus, 262, Holder inequalities, 452 345 holomorphic function, 97 homogeneous solutions (of a Galilean invariance, 423 boundary-value problem), 312 gamma function T(z), 3, 48, 124 hydrodynamics, 448 analytic continuation of, 201 hydrogen bonds, 425 factorial recursion relation for, 4, 58, 59, hyperbolic partial differential equation, 81, 65, 129 167 Laurent series for, 6, 47 hyperbolic velocities, 415 Stirling's formula for, 6 hypercube, 88, 89, 191 as an application of Laplace's method, hypersphere, 183 71-72 hyperspherical coordinates, 91, 187, 189 as an application of the method of hypersurface, 183 steepest descents, 125 gas dynamics, 79 identity matrix, 356 gas particles implicit differentiation, 65, 69, 161, 190, collisions of, 79 407, 409, 412 inertia of, 80 Implicit Function Theorem, 37, 64, 67, 68, gauge functions, 28, 326 77, 83, 92, 156, 188, 190, 287 Gaussian integral, 59 incomplete gamma function j(z,x), 48, 336 evaluation of, 59 large x expansion of, 51, 55 general relativity small x expansion of, 48 Einstein equations of, 78 incompressibility constraint, 80 generating function indicial equation, 219-222 in probability theory, 62 induction (proof by), 283 geometric series, 51, 202, 361, 380 inertia, 80, 176 analytic continuation of, 201, 203 inhomogeneity geometrical optics, see optics, geometrical spatially periodic, 259 global solution, 77, 79, 80, 82, 83 inner asymptotics, 299, 321-325 gradient vector, 98 inner expansion, 11 gravitational acceleration, 434 inner limit, 328 gravitational forces, 79 inner product, 359, 365-368 gravity, 433 Euclidean, 359, 365-367 Green's function on L2(-7r,7r), 378 for the diffusion equation, 82 inner product space, 365 for the Schrodinger equation, 178 inner solution, 298 group velocity, 169-171, 408-410, 412, 415 inner variable, 317, 322 462 Subject Index

nonlinear, 350 in Borel summation theory, 248 integrable system, 310, 447, 449 inverse, 249 integral equations, 399 Laplace transform pair, 135 of Fredholm type, 313 Laplace's equation, 104, 434 system of, 291 Laplace's method, 61-71, 151, 156 integrating factor, 271, 319, 336, 399 application to derivation of Stirling's modified, 284 formula, 71-72 integration by parts, 3, 47, 50, 57, 116, 128, application to shock waves, 73-87 153, 163, 192, 273, 313, 344 for multidimensional integrals, 87—93 applied to oscillatory integrals, 160-161 global versus local maxima in, 84 integration constants, 320, 324-327 use in the method of steepest descents, interatomic force law 98 as a nonlinear spring, 425 lattice points, xiv intermediate limit, 326 counting using the method of stationary intermediate scale, 326, 332 phase, 183-185 specific choice of, 340-342 lattice spacing, 403 intermediate variable, 326 Laurent series, 97 internal layer, 311, 321, 325, 337-339 role in representation of solutions near inverse temperature, 93 singular points, 210-212 inverse-scattering transform, 193, 447, 449 layers (boundary and internal) inviscid fluid, 433 conditions on locations of, 324 irrotational fluid, 433 thickness of, 322-323 isomonodromy theory, 252 leading term, 31, 84, 258, 265, 335, 394 iterates Lebesgue Dominated Convergence sequence of, 232-234 Theorem, 5, 113, 125, 152 iteration, 234 left multiplication, 366 iteration (of integral equations), 232 light cone, 167 light ray, 310 Jacobi elliptic functions, 383 light waves Jacobi polynomials, 73 propagation of, 270 Jacobian, 88, 92, 189, 191, 408 limit cycle, 397 Jordan Normal Form, 363, 364 linear combination, 20, 111, 200, 211, 256, 279, 280, 300, 365, 366, 369, 391, 404, kernel, 358 422 kinetic energy, 426 linear density, 109 Klein-Gordon equation, 165, 448 linear operator, 359, 368 Klein-Gordon equation (nonlinear), 409, densely defined, 367 448 linear space Korteweg-de Vries equation, 13, 416, axioms of, 235—236 424-425, 448 linearized Korteweg-de Vries equation, 165 for the Fermi- Past a- U lam model, little-oh, 17-20, 54, 58, 254, 283 432-433 of one, 19 for water waves, 444-447 relational notation for (

formal expansions for, 371-377 momentum, 426 antiperiodic case of, 370 monatomic chain, 425 justification of formal expansions for, monodromy (about singular points), 377-382 206-213 periodic case of, 370 monodromy matrix, 208, 252 stable case of, 370 as a product of Stokes matrices, 246 unstable case of, 370 diagonalizable, 221 matrix kernel, 291, 309 eigenvalues of, 209, 217, 218, 220-222 matrix norms (induced by £p) invertibility of, 208 equivalence of, 362 left (row) eigenvector of, 209, 211, 218, matter waves 221 dispersion relation for, 171-173 of Floquet theory, 368-370 maximum principle, 104 eigenvalues of, 369 mean flow similarity transforms of, 209 possibility of excitation of, 416, 430 Morse Lemma, 193, 384 rapid development of, 413 multidimensional integrals role of in water wave theory, 439 Laplace's method for, 87-93 spontaneous excitation of, 410-417 the method of stationary phase for, mean term, 405, 411 181-192 Mean Value Theorem, 149, 234 multiple wave resonances, 417-423 mean values, 373 multiplicative identity, 236 method of characteristics, xiv, 75-81, 84, multivaluedness, 127, 210 175 ambiguous predictions of, 84 Navier-Stokes equations, 80 method of Frobenius, 216-222, 249, 251 incompressible, 354 analytical aspect of, 217, 251 nearest-neighbor coupling, 402, 425 formal aspect of, 217 nearly monochromatic waves, 410 use of reduction of order in, 218, 221, 222 negativity condition, 313, 314, 335 method of multiple scales, 400 Neumann series, 361, 362, 380 application of to derivation of universal Newton's equations, 427 equations, 401-425 Newton's first law, 176 for oscillatory problems, 391-397 Newton's method, 234, 235 justification of, 397-400 Newton's second law, 12, 402 in boundary layer theory, 350-353 Nobel Prize, 173 in the context of partial differential Nobel prize, 400 equations, 406 nonlinear coefficient (3 method of stationary phase, 149-164 for the Boussinesq equation, 415 for multidimensional integrals, 181—192 for the sine-Gordon equation, 409 method of steepest descents, 95-108, 151, for water waves, 444 288 incorrect value of, 415 applied to oscillatory integrals, 150, nonlinear lattices, xiv 158-160 nonlinear operator, 232, 272 applied to selection of decaying solutions, modified, 285 142-147 nonlinear parameter /3 method of strained coordinates, see for the Fermi-Pasta-Ulam model, 432 Poincare-Lindstedt method nonlinear Schrodinger equation, 13, 409, method of undetermined coefficients, 352, 410, 413, 415, 448 393 as the wrong model in absence of Minkowski inequalities, 347, 452 dispersion, 432 modal interactions, 422-423 defocusing case of, 410, 417, 432 model equations focusing case of, 410, 417, 432 universal, 13 modulational instability of, 432 modified Korteweg-de Vries equation, 416, for the Fermi-Pasta-Ulam model, 449 427-432 weakly nonlinear, 416 for water waves, 436-444 modulated wavetrains, 169-171 universality class of, 409 moments, 73 nonuniformity, 264 464 Subject Index

norm, 233, 235 partition of unity, 153, 163, 186 axioms of, 236-237 complementary function in, 154 homogeneity of, 236 paths of steepest descent, 98-103 induced by an inner product, 365 global, 101 of a matrix, 362 peptide groups, 425 positive definiteness of, 236 period, 166 triangle inequality for, 237 period of oscillation, 355 normal form periodic solutions, 355 Jordan, see Jordan Normal Form periodic solutions (near equilibrium), of the exponent, 87, 156 383-384 normal forms perturbation problem, 12 analogues of, 14 of linear algebra, 356-368 normal vector, 93, 182 degenerate theory of, 362-365, 375 normed linear space, 233 nondegenerate theory of, 357-361 axioms of, 237 regular, 38-40, 254-263, 293 nullspace, 358, 359 justification of formal expansion for, nullvector, 365 260-263 numerical methods, 36, 52, 53, 120, 315, singular, 13, 40-43, 263 426, 449 rescaling to a regular problem, 43 for root finding, 233-235 phase, 151, 405 Nusselt number, 253 phase integral, 303, 304 phase plane, 384, 395, 397 open covering, 208 phase portrait, 276 optics, 260 phase velocity, 166 geometrical, 310 photoelectric effect, 173 nonlinear, 410 Planck's constant, 172, 175 optimal approximation, 27 Poincare-Hopf Index Theorem, 183 order notation, 15 Poincare-Lindstedt method, 388-391, 406 ordinary points, 200 justification of, 397-400 series solutions about, 215-216 point at infinity, 216, 223 orthogonal polynomials, xiv, 73 Poisson Summation Formula, 183, 184 orthogonal trajectories, 100 polar coordinates, 59 orthogonality, 360, 365 population dynamics, 78 oscillatory integral, 149, 164 potential energy, 426 outer approximation, 11 potential function (for irrotational flow), outer asymptotics, 318-321 434 outer limit, 327 pressure, 80 outer solution, 298 principal axes, 87 outer variable, 318, 322 principle of dominant balance, 40-43, 224, overlap domain, 294, 296, 326 225 choice of scale in, 341-342 for boundary-value problems, 320, 323, properties of, 297-298 325 size in relation to order of matching, 332 in the WKB method, 265 overtones, see harmonics, higher propagation finite speed of, 167 Pade approximation, 147 pseudoinverse, xiv, 359, 361, 378, 379, 382, Painleve equations, 252 400 isomonodromy theory of, 252 construction of using variation of parabolic cylinder functions, 147, 299 parameters, 378 parabolic partial differential equation, 81 paradox,13 quadratic formula, 315 parametric resonance, 260 quanta, 173 and the WKB method, 270 quantum mechanics, xiv, 300, 310 parametrization as a regularization of shock waves, 177 choice of in the method of steepest earliest form of, 304 descents, 107-108 relativistic parity (of functions), 380 of spin-zero particles, 448 Subject Index 465

semiclassical limit of Riccati equation, 224-230, 240, 241, 246, heuristic approach, 174-177 265, 267, 268, 271, 274, 275, 283, 286 rigorous approach, 177-181 application to regular singular points, semiclassical limit of by stationary phase, 241-242 171-181 direction field for, 280 quiescent state, 416 nullclines for, 280 Riemann sense of integration, 21 Riemann sum, 21, 52, 111, 141 radius of convergence, 24, 35, 203, 254 Riemann-Lebesgue Lemma, 152 random matrix theory, xiv, 94 rigid translation, 408, 412 partition function of, 93, 94 RNA molecules range, 358, 359, 365 base-pairs on orthogonal complement of, 367 torsional vibrations of, 402 rank root finding, 36-43, 65, 356 column, 363 numerical methods for, 233-235 equivalence, 363 rule of thumb, 39 row, 363 Rankine-Hugoniot condition, 110 saddle point method, 105 rate of decay saddle points, 103-107 of the error, 23 simple, 105, 113 rational approximation, see Pade saddle-node bifurcation, 179 approximation sawtooth, 260 rational functions, 197 Schrodinger equation, 165, 193 reduced equation, 258, 312, 319, 351, 371, initial data for, 174 403, 406, 411 interpretation of, 173-174 inner, 323, 336 stationary, 300 reduced problem, 254, 256, 258, 293, 356 approximate eigenvalues of, 303 reduction of order, 213-214, 221 dense packing of eigenvalues of, 304 use in the method of Frobenius, 218, 221, eigenfunctions of, 302 222 eigenvalues of, 302 refractive index secular terms, 9, 386-387, 392, 395, 396, modulations in, 270 400, 404-408, 411-413 regular perturbation theory, 264 associated with a resonant mean, 413 regularization, 79 removal of, 388-391, 438 relative error, 23 seed solution, 213, 214, 221 of order e(z), 23 self-adjoint operator, 379 remainder, 6, 38, 47, 50 semiclassical limit, xiv rescaled error, 230 separation constant, 111, 253 differential equation for, 230 separation of variables, 80, 110 integral equation for, 232, 240 series rescaled variable, 11 well-ordered, 283 rescaling, 40-43 sheared graph, 74, 78 in boundary/internal layer theory, shock structure, see shock waves, structure 321-325 of near a turning point, 304 shock time, 78 nonlinear alternative to, 305 shock waves, 73, 78, 108 role in singular asymptotics, 264 regularization of, 78-80 residual, 261, 334, 343, 346-348 structure of, 86 correction to, 343 trajectories of, 85, 86 Residue Theorem, 97, 126, 127, 138, 141 weakly viscous, xiv resonance, 173, 391 analysis of via Laplace's method, accidental, 406, 410, 411 73-87 resonant quartets, 420-422 simultaneous equations, 296 two-parameter family of, 420 sine-Gordon equation, 402, 403, 407, 447, resonant triads, 418-419 448 one-parameter family of, 418 nonlinear coefficient /3 for, 409 Reynolds number, 253 singular asymptotics, 263, 311 466 Subject Index

singular limit, see singular asymptotics square root singular matrix, 365 analytic continuation of, 206 singular perturbation theory, 263 stable manifold, 276, 277 singular points, 146, 200 static phenomena, 355 irregular stationary phase points, 151, 156 asymptotic behavior of solutions near, bifurcation of, 179 223-251 simple, 162 definition of, 212 statistical thermodynamics, 426 definition of (practical), 222 steepest ascent directions, 232 formal series about, 223-229 steepest descent directions (angles), 231, true solutions near, 229-242 240, 241 regular different families of, 231, 241, 242 analysis of via the Riccati equation, steepest descent paths 241-242 global, 106, 114 definition of, 212 Stokes matrix, 244 definition of (practical), 222 for Airy's equation, 245 series solutions about, 216-222 relation to monodromy matrix, 246 regular versus irregular, 206-213 triangular form of, 244 dichotomy defining, 212 Stokes phenomenon, 23, 123, 229, 242-246 slow times, 391, 393, 394 for first-order systems, 251 slowly-varying function, 264, 310 via Borel summation, 251 Sobolev inequality, 345 Stokes rays, 242, 251 solid-state physics, 260, 425 canonical solutions associated with, 243 solitons labeling of, 242 bright, 410 strained coordinate, 388, 391 dark, 410 subsonic frames of reference, 116 of the Korteweg-de Vries equation, 425 subspace of the nonlinear Schrodinger equation, dense, 367, 379 410 of definition for a linear operator, 366 solvability conditions, 358-361, 363, superconducting Josephson junction, 403 365-368, 379, 383, 390, 393, 394, 396, superposition principle, 80, 111 399, 402, 408, 409, 412, 414, 438 supersonic frames of reference, 116 as another eigenvalue problem, 363 surfaces of constant negative curvature, 447 as orthogonality conditions, 365 symbol, 166 for periodic solutions of Mathieu's equation, 373, 374 Taylor polynomials sound speed, 411 zeros of, xiv, 106 spatial period, 259 Taylor series, 4, 6, 8, 24, 26, 37, 38, 48, 87, spatial scales 97, 318 multiple, 406 about a turning point, 287 special functions, 335 of log(l+t2), 58 Airy, see Airy functions of an even function, 59 aspects of, xiv of the exponential function, 106 classical, 123 remainder term in, 25, 53, 54, 65, 89, 254 gamma, see gamma function use in analytic continuation, 201-203 Hankel, see Hankel function Taylor's Theorem, 25, 68, 156, 157 incomplete gamma, see incomplete test function, 183 gamma function thermalization, 426 Jacobi elliptic, see Jacobi elliptic threshold distance (for the WKB method functions near turning points), 283 parabolic cylinder, see parabolic cylinder topography, 434, 449 functions total degree, 424 spectral decomposition, 358 total energy spectral gap of the Fermi-Pasta-Ulam model, 426 for the Klein-Gordon equation, 167 tracking stable and unstable manifolds, 277 spectral radius, 362 transcendentally small term, 30 spring constant, 402 transform, 135 Subject Index 467

transform integrals generalizations of, 56-60, 69 asymptotics of, 135-142 wave equation, 165 transitional phenomena, 315 d'Alembert formula for, see d'Alembert transitive property, 20 formula triangle inequality, 26, 34, 237, 348, 451 nonlinear, 402 generalizations of, 452 wave function, 174 triple-valued function, 78 phase of, 403 triple-valued profile, 74 wave mechanics, 13 turbulent flows, 354 wave mixing, 418 turning points, 277-310 wave of depression, 436 higher-order, 278 wave of elevation, 436 multiple, 300-304 wavelength, 166, 259, 270 problems associated with, 278-280 wavenumber, 166 simple, 277, 280, 283, 298, 301, 305 waves asymptotics near, 287-293 propagation of in an inhomogeneous sneaking up on with the WKB method, medium, 258 283-287 weak cubic nonlinearity, 398 perturbative approach to, 384-388, undisturbed depth, 433 391-394 uniform convergence, 380 weak damping, 395-397 uniform validity, 8, 11, 264, 390 weak quadratic nonlinearity uniformly valid approximation, 327 perturbative approach to, 390 unique characteristic selection principle, 86, weakly nonlinear oscillations, 382-400 177 Weber's equation, 147, 197, 299, 300 variational character of, 86 perturbation of, 299, 310 unit vector well-ordering (of terms in a series), 283 standard, 359 WKB method, 283, 310, 311, 355, 371 unitary ensemble, 93 application to Mathieu's equation, 377 universal amplitude equation, 409 applied to optics, 270 universality, 13, 401-425 directional character of approximations, unstable manifold, 276 275-277, 280 upper limit of integration failure of near turning points, 278 role in Watson's Lemma, 55 general, 263-268 nonlinear analogues of, 310 variation of parameters, 256-260, 290, 308, turning points in, 277-310 324, 352, 372, 374, 385, 387, 389 validity of (away from turning points), for first-order systems, 399 270-275 use in constructing a pseudoinverse, 378 when coefficients have asymptotic power variational principle, 353 series, 268-270 vibrational modes, 426 WKBJ method, 310 viscosity, 80 Wronskian, 243, 257, 259, 369 vortex shedding, 354 constancy of, 369 of Ai(z) and Bi(z), 289, 294 water wave problem use in detection of eigenvalues, 302 bottom boundary condition for, 435 nondimensionalization of, 434-435 zero element, 235 top boundary conditions for, 436 zero-diffusion limit, 84-86 water waves, xiv, 433-447, 449 zero-frequency term, 391 dispersion relation for, 438 Korteweg-de Vries equation for, 444-447 nondimensional measure of, 435 nonlinear coefficient (3 for, 444 nonlinear Schrodinger equation for, 436-444 wavelength stability threshold for, 444 Watson's Lemma, 52-56, 61, 64, 65, 67, 69, 130, 139, 142, 145, 146, 248, 251 This page intentionally left blank Titles in This Series

75 Peter D. Miller, Applied asymptotic analysis, 2006 74 V. V. Prasolov, Elements of combinatorial and differential topology, 2006 70 Sean Dineen, Probability theory in finance, 2005 69 Sebastian Montiel and Antonio Ros, Curves and surfaces, 2005 68 Luis Caffarelli and Sandro Salsa, A geometric approach to free boundary problems, 2005 67 T.Y. Lam, Introduction to quadratic forms over fields, 2004 66 Yuli Eidelman, Vitali Milman, and Antonis Tsolomitis, Functional analysis, An introduction, 2004 65 S. Ramanan, Global calculus, 2004 64 A. A. Kirillov, Lectures on the orbit method, 2004 63 Steven Dale Cutkosky, Resolution of singularities, 2004 62 T. W. Korner, A companion to analysis: A second first and first second course in analysis, 2004 61 Thomas A. Ivey and J. M. Landsberg, Cartan for beginners: Differential geometry via moving frames and exterior differential systems, 2003 60 Alberto Candel and Lawrence Conlon, Foliations II, 2003 59 Steven H. Weintraub, Representation theory of finite groups: algebra and arithmetic, 2003 58 Cedric Villani, Topics in optimal transportation, 2003 57 Robert Plato, Concise numerical mathematics, 2003 56 E. B. Vinberg, A course in algebra, 2003 55 C. Herbert Clemens, A scrapbook of complex curve theory, second edition, 2003 54 Alexander Barvinok, A course in convexity, 2002 53 Henryk Iwaniec, Spectral methods of automorphic forms, 2002 52 Ilka Agricola and Thomas Friedrich, Global analysis: Differential forms in analysis, geometry and physics, 2002 51 Y. A. Abramovich and C. D. Aliprantis, Problems in operator theory, 2002 50 Y. A. Abramovich and C. D. Aliprantis, An invitation to operator theory, 2002 49 John R. Harper, Secondary cohomology operations, 2002 48 Y. Eliashberg and N. Mishachev, Introduction to the ^-principle, 2002 47 A. Yu. Kitaev, A. H. Shen, and M. N. Vyalyi, Classical and quantum computation, 2002 46 Joseph L. Taylor, Several complex variables with connections to algebraic geometry and Lie groups, 2002 45 Inder K. Rana, An introduction to measure and integration, second edition, 2002 44 Jim Agler and John E. McCarthy, Pick interpolation and Hilbert function spaces, 2002 43 N. V. Krylov, Introduction to the theory of random processes, 2002 42 Jin Hong and Seok-Jin Kang, Introduction to quantum groups and crystal bases, 2002 41 Georgi V. Smirnov, Introduction to the theory of differential inclusions, 2002 40 Robert E. Greene and Steven G. Krantz, Function theory of one complex variable, third edition, 2006 39 Larry C. Grove, Classical groups and geometric algebra, 2002 38 Elton P. Hsu, Stochastic analysis on manifolds, 2002 37 Hershel M. Farkas and Irwin Kra, Theta constants, Riemann surfaces and the modular group, 2001 36 Martin Schechter, Principles of functional analysis, second edition, 2002 35 James F. Davis and Paul Kirk, Lecture notes in algebraic topology, 2001 TITLES IN THIS SERIES

34 Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, 2001 33 Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, 2001 32 Robert G. Bartle, A modern theory of integration, 2001 31 Ralf Korn and Elke Korn, Option pricing and portfolio optimization: Modern methods of financial mathematics, 2001 30 J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, 2001 29 Javier Duoandikoetxea, Fourier analysis, 2001 28 Liviu I. Nicolaescu, Notes on Seiberg-Witten theory, 2000 27 Thierry Aubin, A course in differential geometry, 2001 26 Rolf Berndt, An introduction to symplectic geometry, 2001 25 Thomas Friedrich, Dirac operators in Riemannian geometry, 2000 24 Helmut Koch, Number theory: Algebraic numbers and functions, 2000 23 Alberto Candel and Lawrence Conlon, Foliations I, 2000 22 Giinter R. Krause and Thomas H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, 2000 21 John B. Conway, A course in operator theory, 2000 20 Robert E. Gompf and Andras I. Stipsicz, 4-manifolds and Kirby calculus, 1999 19 Lawrence C. Evans, Partial differential equations, 1998 18 Winfried Just and Martin Weese, Discovering modern set theory. II: Set-theoretic tools for every mathematician, 1997 17 Henryk Iwaniec, Topics in classical automorphic forms, 1997 16 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume II: Advanced theory, 1997 15 Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Volume I: Elementary theory, 1997 14 Elliott H. Lieb and Michael Loss, Analysis, 1997 13 Paul C. Shields, The ergodic theory of discrete sample paths, 1996 12 N. V. Krylov, Lectures on elliptic and parabolic equations in Holder spaces, 1996 11 Jacques Dixmier, Enveloping algebras, 1996 Printing 10 Barry Simon, Representations of finite and compact groups, 1996 9 Dino Lorenzini, An invitation to arithmetic geometry, 1996 8 Winfried Just and Martin Weese, Discovering modern set theory. I: The basics, 1996 7 Gerald J. Janusz, Algebraic number fields, second edition, 1996 6 Jens Carsten Jantzen, Lectures on quantum groups, 1996 5 Rick Miranda, Algebraic curves and Riemann surfaces, 1995 4 Russell A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, 1994 3 William W. Adams and Philippe Loustaunau, An introduction to Grobner bases, 1994 2 Jack Graver, Brigitte Servatius, and Herman Servatius, Combinatorial rigidity, 1993 1 Ethan Akin, The general topology of dynamical systems, 1993