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Textbook:
Lecture 1
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HISTORICAL INTRODUCTION:
Early observations of nonlinear coherent structures in
energy conserving systems, solitons. Nonlinear diffusion.
NLS: 1.1, 1.2, 1.3, 1.4
Lecture 2
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LINEAR DISPERSIVE SYSTEMS. LINEAR DIFFUSION:
Laplace and Fourier transforms. Conservation theorems.
NLS: 2.1, 2.2, 2.3
Lecture 3
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DRIVEN SYSTEMS. LINEAR SCATTERING THEORY
Green's method, Fredholm's
theorem. Linear stability. Schrödinger's equation. Gelfand-Levitan theory.
Reflectionless potentials. NLS: 2.4, 2.5, 2.6
Lecture 4
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CLASSICAL SOLITON EQUATIONS:
KORTEWEG deVRIES EQUATION (KdV)
Shallow water waves. Fermi-Pasta-Ulam chain: continuum
approximation
NLS: 3.1.1 + notes
Lecture 5
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SOLUTION TECHNIQUES: Travelling waves. Solitary waves. Periodic
waves.
BÄCKLUND TRANSFORMATION. HIROTA'S METHOD.
NLS: 3.1.2 - 3.1.5
Lecture 6
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SINE-GORDON EQUATION (sG)
Long Josephson junctions. Dislocation theory. Mechanical
analogue.
NLS: 3.2.1-3.2.2 + notes
Lecture 7
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SOLUTION TECHNIQUES:
BÄCKLUND TRANSFORMATION. LAMB'S METHOD.
NLS: 3.2.3 - 3.2.6
COMPUTER MOVIE SINE-GORDON SOLITONS
Lecture 8
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INVERSE SCATTERING THEORY
Linear scattering revisited. Inverse scattering method
for KdV
NLS: 6.1 + notes
Lecture 9
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INVERSE SCATTERING THEORY
Inverse scattering theory for KdV, sG, and NLS
NLS: 6.1, 6.2, and briefly 6.3-5.
Lecture 10
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NONLINEAR DIFFUSION IN EXCITABLE MEDIA:
A simple example: the candle. Propagation of a chemical
reaction.
NLS: 4.1.1 and 4.1.2
Lecture 11
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THE HODGKIN-HUXLEY SYSTEM
Properties of the nerve cell and its electrodynamics.
Excitable media
NLS: 4.2
Lecture 12
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THE FITZHUGH-NAGUMO NERVE.
Stability. Neural novelties
NLS: 4.3, 4.4
Lecture 13
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NONLINEAR-SCHRÖDINGER EQUATION (NLS):
Wave packet. Kerr media. Derivation from Maxwell's equations.
NLS: 3.3.1
Lecture 14
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SOLUTION TECHNIQUES AND APPLICATIONS:
X(2)media. Single soliton.
Bäcklund transformation. Periodic solutions. Symmetries and
conservation theorems. Computer examples. Vector NLS equation. Collapse
NLS: 3.3.2 - 3.3.4 + note
Lecture 15
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NONLINEAR SCHRÖDINGER LATTICES
- Discrete Nonlinear Schrödinger and Ablowitz-Ladik
lattices. Poisson brackets.
NLS: 5.3 + note
Lecture 16
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BASIC NONLINEAR LATTICES.
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Toda Lattice. Nonintegrable Lattice
NLS: 5.1.1 - 5.1.3
Lecture 17
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SOLITONS IN BIOMOLECULES:
Stability of DST equation
Biological solitons.
Perturbed matrices
NLS: 5.4.0 + 5.4.1 + note
(Lagrange and Hamilton function for
DST equation)
Lecture 18
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SOLITONS IN BIOMOLECULES:
Stability of DST equation
Biological solitons.
Perturbed matrices
NLS: 5.4.2, 5.6, 7.1
Lecture 19
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PERTURBATION THEORY:
Damped harmonic oscillator
Multiple time scales
Matematica Notebooks (1a-b) handed out
NLS: 7.2.1, 7.2.2 + Mathematica Notebook
Lecture 20
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ENERGY ANALYSIS OF SOLITON DYNAMICS:
KdV solitons.
Fluxons in Josephson junctions.
Mathematica Notebook
NLS: 7.3.1, 7.3.2
Lecture 21
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ENERGY ANALYSIS OF SOLITON DYNAMICS:
Light pulses: nonlinear Schrödinger solitons
NLS: 7.3.3
Lecture 22
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FROM CLASSICAL TO QUANTUM OSCILLATORS
The birth of quantum theory (8.1.2).
A quantum linear oscillator (8.1.3).
The rotating wave approximation (8.1.4)
Note on the relation between classical and quantum mechanics
Lecture 23
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QUANTUM OSCILLATORS
Recapitulation of quantization (8.1.3-8.1.4).
The Born-Oppenheimer approximation briefly (8.1.5).
Dirac's notation (8.1.6).
NLS: 8.1 + note
Lecture 24
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SELF-TRAPPING IN BIOMOLECULES
CH stretching oscillations in methanes.
NLS: 8.2
Lecture 25
BOSON LATTICES
Example of CH stretching oscillations (f=2, n=3).
Quantization of the discrete nonlinear self-trapping
equation.
Briefly on translational invariance in a lattice nonlinear
Schrödinger equation.
NLS: 8.3 (till page top of page 387).
Briefly on fermionic models (p. 400).
Lecture 26
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OVERVIEW
Summarizing the contents of the course.
Perspectives.
For further information, please contact plc, IMM
Bldg. 305, DTU
Phone: +45 45253092 / Fax: +45 45931235, E-mail: plc@imm.dtu.dk.