# Contents of Lectures

• Textbook:

• Lecture 1

• 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
• LINEAR DISPERSIVE SYSTEMS. LINEAR DIFFUSION:

• Laplace and Fourier transforms. Conservation theorems.
NLS: 2.1, 2.2, 2.3
Lecture 3
• 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
• CLASSICAL SOLITON EQUATIONS:

• KORTEWEG deVRIES EQUATION (KdV)
Shallow water waves. Fermi-Pasta-Ulam chain: continuum approximation
NLS: 3.1.1 + notes
Lecture 5
• SOLUTION TECHNIQUES: Travelling waves. Solitary waves. Periodic waves.

• BÄCKLUND TRANSFORMATION. HIROTA'S METHOD.
NLS: 3.1.2 - 3.1.5
Lecture 6
• SINE-GORDON EQUATION (sG)

• Long Josephson junctions. Dislocation theory. Mechanical analogue.
NLS: 3.2.1-3.2.2 + notes
Lecture 7
• SOLUTION TECHNIQUES:

• BÄCKLUND TRANSFORMATION. LAMB'S METHOD.
NLS: 3.2.3 - 3.2.6
COMPUTER MOVIE SINE-GORDON SOLITONS
Lecture 8
• INVERSE SCATTERING THEORY

• Linear scattering revisited. Inverse scattering method for KdV
NLS: 6.1 + notes
Lecture 9
• INVERSE SCATTERING THEORY

• Inverse scattering theory for KdV, sG, and NLS
NLS: 6.1, 6.2, and briefly 6.3-5.
Lecture 10
• 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
• THE HODGKIN-HUXLEY SYSTEM

• Properties of the nerve cell and its electrodynamics. Excitable media
NLS: 4.2
Lecture 12
• THE FITZHUGH-NAGUMO NERVE.

• Stability. Neural novelties
NLS: 4.3, 4.4
Lecture 13
• NONLINEAR-SCHRÖDINGER EQUATION (NLS):

• Wave packet. Kerr media. Derivation from Maxwell's equations.
NLS: 3.3.1
Lecture 14
• 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
• NONLINEAR SCHRÖDINGER LATTICES
• Discrete Nonlinear Schrödinger and Ablowitz-Ladik lattices. Poisson brackets.
NLS: 5.3 + note
Lecture 16
• BASIC NONLINEAR LATTICES.
• Toda Lattice. Nonintegrable Lattice

• NLS: 5.1.1 - 5.1.3
Lecture 17
• 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
• SOLITONS IN BIOMOLECULES:

• Stability of DST equation
Biological solitons.
Perturbed matrices
NLS: 5.4.2, 5.6, 7.1
Lecture 19
• 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
• ENERGY ANALYSIS OF SOLITON DYNAMICS:

• KdV solitons.
Fluxons in Josephson junctions.
Mathematica Notebook
NLS: 7.3.1, 7.3.2
Lecture 21
• ENERGY ANALYSIS OF SOLITON DYNAMICS:

• Light pulses: nonlinear Schrödinger solitons
NLS: 7.3.3
Lecture 22
• 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
• 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
• 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
• OVERVIEW

• Summarizing the contents of the course.
Perspectives.