本书是一本物理科学家的著作，其语言是奇异摄动分析。本书对研究常微分方程（ODE）和偏微分方程（PDE）边值问题的奇异摄动分析的数学家和物理学家非常有用。阅读本书需具备基本的分析技能和良好的数学物理课程基础。 This book is the testimony of a physical scientist whose language is singular perturbation analysis. Classical mathematical notions, such as matched asymptotic expansions, projections of large dynamical systems onto small center manifolds, and modulation theory of oscillations based either on multiple scales or on averaging/transformation theory, are included. The narratives of these topics are carried by physical examples: Let’s say that the moment when we “see” how a mathematical pattern fits a physical problem is like “hitting the ball”. Yes, we want to hit the ball. But a powerful stroke includes the follow-through. One intention of this book is to discern in the structure and/or solutions of the equations their geometric and physical content. Through analysis, we come to sense directly the shape and feel of phenomena.

前辅文
Acknowledgments
Introduction
Chapter 1. What is a singular perturbation?
  Prototypical examples
   Singularly perturbed polynomial equations
   Radiation reaction
   Convection-diffusion boundary layer
   Modulated oscillations
  Guide to bibliography
Chapter 2. Asymptotic expansions
  Problem 2.1: Uniqueness
  A divergent but asymptotic series
   Problem 2.2: Divergent outer expansion
   Problem 2.3: Another outrageous example
  Asymptotic expansions of integrals — the usual suspects
   Problem 2.4: Simple endpoint examples
   Problem 2.5: Stirling approximation to n!
   Problem 2.6: Endpoint and minimum both contribute
   Problem 2.7: Central limit theorem
  Steepest descent method
  Chasing the waves with velocity v >0
  No waves for v <0
   Problem 2.8: Steepest descent asymptotics
  A primer on linear waves
   Problem 2.9: Amplitude transport
   Problem 2.10: How far was that meteor?
   Problem 2.11: Wave asymptotics in non-uniform medium
  A hard logarithmic expansion
   Problem 2.12: Logarithmic expansion
  Guide to bibliography
Chapter 3. Matched asymptotic expansions
  Problem 3.1: Physical scaling analysis of boundary layer thickness
  Problem 3.2: Higher-order matching
  Problem 3.3: Absorbing boundary condition
  Matched asymptotic expansions in practice
   Problem 3.4: Derivative layer
  Corner layers and internal layers
   Problem 3.5: Phase diagram
   Problem 3.6: Internal derivative layer
   Problem 3.7: Where does the kink go?
  Guide to bibliography
Chapter 4. Matched asymptotic expansions in PDE’s
  Moving internal layers
  Chapman–Enskog asymptotics
   Problem 4.1: Relaxation of kink position
   Problem 4.2: Hamilton–Jacobi equation from front motion
   Problem 4.3: Chapman–Enskog asymptotics
  Projected Lagrangian
   Problem 4.4: Circular fronts in nonlinear wave equation
   Problem 4.5: Solitary wave dynamics in two dimensions
   Problem 4.6: Solitary wave diffraction
  Singularly perturbed eigenvalue problem
  Homogenization of swiss cheese
   Problem 4.7: Neumann boundary conditions and effective dipoles
   Problem 4.8: Two dimensions
  Guide to bibliography
Chapter 5. Prandtl boundary layer theory
  Stream function and vorticity
  Preliminary non-dimensionalization
  Outer expansion and “dry water”
  Inner expansion
   Problem 5.1: Vector calculus of boundary layer coordinates
  Leading order matching and a first integral
   Problem 5.2: The body surface is a source of vorticity
   Problem 5.3: Downstream evolution
  Displacement thickness
  Solutions based on scaling symmetry
  Blasius flow over flat plate
  Nonzero wedge angles (m≠0)
  Precursor of boundary layer separation
   Problem 5.4: Wedge flows with source
   Problem 5.5: Mixing by vortex
  Guide to bibliography
Chapter 6. Modulated oscillations
  Physical flavors of modulated oscillations
   Problem 6.1: Beats
   Problem 6.2: The beat goes on
   Problem 6.3: Wave packets as beats in spacetime
   Problem 6.4: Adiabatic invariant of harmonic oscillator
   Problem 6.5: Passage through resonance for harmonic oscillator
   Problem 6.6: Internal resonance between waves on a ring
  Method of two scales
   Problem 6.7: Nonlinear parametric resonance
   Problem 6.8: Forced van der Pol ODE
   Problem 6.9: Inverted pendulum
  Strongly nonlinear oscillations and action
   Problem 6.10: Energy, action and frequency
   Problem 6.11: Hamiltonian analysis of the adiabatic invariant
   Problem 6.12: Poincar´e analysis of nonlinear oscillations
  A primer on nonlinear waves
  Modulation Lagrangian
   Problem 6.13: Nonlinear geometric attenuation
   Problem 6.14: Modulational instability
  A primer on homogenization theory
   Problem 6.15: Direct homogenization
  Guide to bibliography
Chapter 7. Modulation theory by transforming variables
  Transformations in classical mechanics
   Problem 7.1: Geometry of action-angle variables
   Problem 7.2: Stokes expansion for quadratically nonlinear oscillator
   Problem 7.3: Frequency-action relation
   Problem 7.4: Follow the bouncing ball
  Near-identity transformations
   Problem 7.5: van der Pol ODE by near-identity transformations
   Problem 7.6: Subtle balance between positive and negative damping
   Problem 7.7: Adiabatic invariants again
  Dissipative perturbations of the Kepler problem
  Modulation theory of damped orbits
  Guide to bibliography
Chapter 8. Nonlinear resonance
  Problem 8.1: Modulation theory of resonance
  A prototype example
  What resonance looks like
   Problem 8.2: Resonance of the bouncing ball
   Problem 8.3: Resonance by rebounds off a vibrating wall
  Generalized resonance
  Energy beats
  Modulation theory of generalized resonance
   Problem 8.4: Modulation theory for generalized resonance
  Thickness of the resonance annulus
  Asymptotic isolation of resonances
  Guide to bibliography
Bibliography
Index