目錄
Acknowledgments
Introduction
Chapter 1. What is a singular perturbation
Prototypical examples
Singularly perturbed polynomial equations
Radiation reaction
Problem 1.1 : Bad truncations
Problem 1.2 : Harmonic oscillator with memory, and even worse truncations
Convection-diffusion boundary layer
Problem 1.3 : A simple boundary layer
Problem 1.4 : Pileup near x
Modulated oscillationsi etg
Problem 1.5 : Secular terms
Problem 1.6 : Approach to limit cycle
Problem 1.7 : Adiabatic invariant for particle in a box
Guide to bibliography
Chapter 2. Asymptotic expansions
Problem 2.1 : Uniqueness
A divergent but asymptotic seriesyb
Problem 2.2 : Divergent outer expansion volle
Problem 2.3 : Another outrageous example
Asymptotic expansions of integrals - the usual suspects
Problem 2.4 : Simple endpoint exampleswsk
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>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 problemo TotonA:8.9 msldors
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-olioe
Preliminary non-dimensionalization iermqo undo
Outer expansion and 「dry water」
Inner expansion
Problem 5.1 : Vector calculus of boundary layer coordinates
Leading order matching and a first integrals
Problem 5.2 : The body surface is a source of vorticity
Problem 5.3 : Downstream evolutionisd oltca
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 dho
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 gargoildid
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? 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 variable
Transformations in classical mechanics
Problem 7.1 : Geometry of action-angle variables noiouu mso
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 sioi u? y bio
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
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