Applied Asymptotic Analysis

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1 Applied Asymptotic Analysis Peter D. Miller Graduate Studies in Mathematics Volume 75 American Mathematical Society Providence, Rhode Island

2 Preface xiii Part 1. Fundamentals Chapter 0. Themes of Asymptotic Analysis Theme: Asymptotics, Convergent and Divergent Asymptotic Series Theme: Other Parameters and Nonuniformity First example. Oscillations Second example. Boundary layers Theme: Differential Equations Theme: Universal Partial Differential Equations and Canonical Physical Models 13 Chapter 1. The Nature of Asymptotic Approximations Asymptotic Approximations and Errors Order relations among functions Statements following from the order relations Absolute and relative errors Convergent versus Asymptotic Series: Concepts Convergent power series ' Introduction to asymptotic series Asymptotic Sequences and Series: General Definitions How to "Sum" an Asymptotic Series Asymptotic Root Finding A regular perturbation problem 38 vii

3 viii Contents A singular perturbation problem. Rescaling and the principle of dominant balance Notes and References 43 Part 2. Asymptotic Analysis of Exponential Integrals Chapter 2. Fundamental Techniques for Integrals Review of Basic Methods Exponential Integrals and Watson's Lemma Elementary Generalizations of Watson's Lemma 56 Chapter 3. Laplace's Method for Asymptotic Expansions of Integrals Introduction Nonlocal Contributions Contributions from Endpoints Contributions from Interior Maxima Summary of Generic Leading-order Behavior Application: Weakly Diffusive Regularization of Shock Waves The method of characteristics Regularization of shocks by diffusion. Burgers' equation The Cole-Hopf transformation and the solution of the initial-value problem for Burgers' equation Analysis of the solution in the limit of vanishing diffusion Multidimensional Integrals Notes and References 93 Chapter 4. The Method of Steepest Descents for Asymptotic Expansions of Integrals Introduction Contour Deformation Paths of Steepest Descent Saddle Points Parametrization-independent Local Contributions Application: Long-time Asymptotic Behavior of Diffusion Processes A derivation of the diffusion equation 109

4 ix Solution of the diffusion equation and the corresponding initial-value problem Long-time asymptotics via the method of steepest descents Application: Asymptotic Behavior of Special Functions, Airy Functions and the Stokes Phenomenon Integral representations for Airy functions Preliminary transformations necessary for asymptotic analysis of Ai(x) for large x Determination of the path. Dependence of the path on K Asymptotic behavior of Ai(x) for large x. The Stokes phenomenon The Effect of Branch Points Application: Asymptotics of transform integrals Application: Selection of particular solutions of linear differential equations admitting integral representations Notes and References 147 Chapter 5. The Method of Stationary Phase for Asymptotic Analysis of Oscillatory Integrals Introduction Nonlocal Contributions Contributions from Interior Stationary Phase Points Putting the exponent in normal form by a change of variables Analysis of Ji(A) by the method of steepest descents Analysis of J2(A) using integration by parts The asymptotic contribution of a stationary phase point Summary of Generic Leading-order Behavior Application: Long-time Behavior of Linear Dispersive Waves Partial differential equations for linear dispersive waves Analysis of the solution formula. Longtime asymptotics using the method of stationary phase Structure of the wave field for large time. Modulated wavetrains and group velocity Application: Semiclassical Dynamics of Free Particles in Quantum Mechanics Derivation of the dispersion relation for "matter waves" 171

5 The Schrodinger equation for a free particle. Interpretation of the Schrodinger wave function The semiclassical limit. Heuristic reasoning Rigorous semiclassical asymptotics using the method of stationary phase Multidimensional Integrals Notes and References 193 Part 3. Asymptotic Analysis of Differential Equations Chapter 6. Asymptotic Behavior of Solutions of Linear Secondorder Differential Equations in the Complex Plane Qualitative Theory of Solutions Reduction to canonical form Solutions viewed as analytic functions of the complex variable z Reduction of order Asymptotic Behavior near Ordinary and Regular Singular Points Series solutions at ordinary points Series solutions at regular singular points. The method of Frobenius Asymptotic Behavior near Irregular Singular Points Formal asymptotic series Existence of true solutions described by the formal asymptotic series. The Stokes phenomenon Another approach to the existence of true solutions and the Stokes phenomenon. Borel summation Notes and References 251 Chapter 7. Introduction to Asymptotics of Solutions of Ordinary Differential Equations with Respect to Parameters Regular Perturbation Problems, Formal power series expansions Solving for y n (x). Variation of parameters Justification of the formal expansion Singular Asymptotics The WKB method The special case of an asymptotic power series for f(x;\) Turning points 277

6 xi Problems with more than one turning point. The Bohr- Sommerfeld quantization rule Uniform asymptotics near turning points. Langer transformations Notes and References. 310 Chapter 8. Asymptotics of Linear Boundary-value Problems Asymptotic Existence of Solutions Case I: a(x) ^ 0 on [ct,/3] and e is positive but sufficiently small Case II: b(x) - a'(x)/2 < 0 on [a,/3] and e is positive An Exactly Solvable Boundary-value Problem: Phenomenology of Boundary Layers Outer Asymptotics Rescaling and Inner Asymptotics for Boundary Layers and Internal Layers Matching of Asymptotic Expansions, Intermediate Variables, and Uniformly Valid Asymptotics Examples Proving the Validity of Uniform Approximations The Method of Multiple Scales Notes and References 353 Chapter 9. Asymptotics of Oscillatory Phenomena Perturbation Theory in Linear Algebra and Eigenvalue Problems Nondegenerate theory Degenerate theory More on solvability conditions. Inner products and adjoints Periodic Boundary Conditions and Mathieu's Equation Floquet theory Periodic and antiperiodic solutions. Formal asymptotics Justification of the expansions Weakly Nonlinear Oscillations Periodic solutions near equilibrium A perturbative approach to weak cubic nonlinearity. Secular terms Removal of secular terms. Strained coordinates and the Poincare-Lindstedt method 388

7 xii Contents The method of multiple scales Justification of the expansions Notes and References 400 Chapter 10. Weakly Nonlinear Waves Derivation of Universal Partial Differential Equations Using the Method of Multiple Scales Modulated wavetrains with dispersion and nonlinear effects. The cubic nonlinear Schrodinger equation Spontaneous excitation of a mean flow Multiple wave resonances Long wave asymptotics. The Boussinesq equation and the Korteweg-de Vries equation Waves in Molecular Chains The Fermi-Pasta-Ulam model Derivation of the cubic nonlinear Schrodinger equation Derivation of the Boussinesq and Kortewegde Vries equations Water Waves Derivation of the cubic nonlinear Schrodinger equation Derivation of the Korteweg-de Vries equation 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

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