04225 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS, SOLITONS

Content of Lectures

Textbook:

NONLINEAR SCIENCE: Emergence and Dynamics of Coherent Structures, Alwyn Scott, with contributions by M. P. Sørensen, and P. L. Christiansen. Oxford University Press, 1999. (NLS).

Lecture 1

HISTORICAL INTRODUCTION: Nonlinear copherent structures in energy conserving systems, solitons, transformation methods.

NLS: 1.1, 1.2, 1.3, 1.4.

Lecture 2

LINEAR DISPERSIVE SYSTEMS: Green's method, Fredholm's theorem, scattering theory.

NLS: 2.1, 2.4, 2.6.

Lecture 3

THE FERMI-PASTA-ULAM PROBLEM AND THE KORTEWEG-DEVRIES EQUATION:
Introduction to solitons.
Note 1 by Michel Peyrard (available at Room 017, Building 305)

Lecture 4

SOME PROPERTIES OF THE KDV EQUATION;
Solitons and multisolitons.
Examples: plasmas, electrical lines, blood-pressure waves.
NLS: 3.1.1. and pp. 70-71.
M. Remoissenet, "Waves Called Solitons", Springer (1994): Sections 3.2 and 3.3

Lecture 5

DISLOCATION IN CRYSTALS AND THE SINE-GORDON MODEL
AND ITS ENERGY LANDSCAPE:
Introduction to topological solitons
Note 2 by Michel Peyrard (available at Room 017, Building 305)
NLS: pp. 79-80

Lecture 6

PROPERTIES AND APPLICATIONS OF THE SINE-GORDON EQUATION:
Energy of a soliton, multisolitons and breathers. Other nonlinear Klein-Gordon equations.
M. Remoissenet, "Waves Called Solitons", Springer (1994): Sections 6.1-6.5.

Lecture 7

INCOMMENSURATE PHASES:
An application of the sine-Gordon equation, and a link with nonlinear dynamical systems.
Note 3 by Michel Peyrard (available at Room 017, Building 305)

Lecture 8

INCOMMENSURATE PHASES: conclusion
FERROELECTRIC DOMAIN WALLS:
Introduction to ferroelectrics. Domain walls as solitons.
Dielectric response of a ferroelectric crystal.
Note 4 by Michel Peyrard (available at Room 017, Building 305)
Also: J. A. Krumhansl and J. R. Schrieffer, Physical Review B 11 (1975) 3535

Lecture 9

CONDUCTING POLYMERS: (Chemistry Nobel Prize 2000)
Electrical conduction by charged solitons. The infra-red response: collective coordinate description.
Note 5 by Michel Peyrard (available at Room 017, Building 305)

Lecture 10

NONLINEAR DYNAMICS OF DNA:
Nonlinear localization and DNA, breathing and melting.
Note 6 by Michel Peyrard (available at Room 017, Building 305)

Lecture 11

THE NONLINEAR SCHRÖDINGER EQUATION:
Multiple scale perturbation theory for weakly nonlinear hyperbolic quations.
The nonlinear Schrödinger equation.
Single simple soliton solution.

NLS: pp. 97-99.

Lecture 12

THE NONLINEAR SCHRÖDINGER EQUATION
AND TRANSVERSE PHENOMENA:
Nonlinear wave packets.
Bäcklund transformation (also for sine-Gordon equation)
A vector NLS equation, collapse and blow-up, filamentation.

NLS: 3.3.1 (pp. 94-97), 3.3.3 (and p. 7), 3.3.4.

Lecture 13

INVERSE SCATTERING THEORY (IST):
The Inverse Scattering Transform method. Lax pair for KdV equation.
Three steps: 1) Initial forward scattering. 2) Time dependance af scattering data, and 3) Final inverse problem.
Delta potential.
Scattering data at time t.
NLS: pp 241-244 and 6.2.1. Recapitulation of pp. 42-46.

Lecture 14

INVERSE SCATTERING THEORY:
Reflectionless potential.
Gelfand-Levitan theory.
Solving the KdV equation and other integrable systems by IST.
Note by Peter L. Christiansen (available at Room 017, Building 305)
NLS: 2.6.2 and 6.2.2

Lecture 15

LINEAR DIFFUSION:
Fourier and Laplace transforms, Green's method.

NLS: 2.3, 2.4.1.

Lecture 16

NONLINEAR DIFFUSION:
Phase plane analysis of simple traveling waves.
NLS: 4.1.1.

Lecture 17

REAL NERVES:
Qualitative switching behavior of a nerve membrane, the Hodgkin-Huxley formulation,
estimates of the traveling wave speed, electrical circuit equivalent.

NLS: 4.2.

Lecture 18

THE FITZHUGH-NAGUMO EQUATION:
Traveling wave analysis, qualitative behavior,
the power balance condition.

NLS: 4.3.

Lecture 19

DECREMENTAL CONDUCTION:
Qualitative effects of temperature, ionic concentrations,
and the refractory zone on the dynamics of a nerve impulse.

NLS: 4.5.1.

Lecture 20

THRESHOLD:
Conditions for launching a nerve impulse, effects of tapering and varicosities.
NLS: 4.5.2, 4.5.3.

Lecture 21

DENDRITIC LOGIC:
The geometric ratio, action potentials on dendrites, dendritic logic.

NLS: 4.5.5.

Lecture 22

ELEMENTARY PERTURBATION THEORY:
Eigenvalues of perturbed matrices, the damped harmonic oscillator. Power series expansions,
elimination of secularities.
NLS: 7.1 and 7.2.

Lecture 23

ENERGY ANALYSES OF SOLITON DYNAMICS:
Effects of dissipation on dynamics of Korteweg de Vries, nonlinear Schroedinger
and sine-Gordon solitons.

NLS: 7.3, 7.4.

Lecture 24

MULTISOLITON PERTURBATION THEORY:
Fredholm's theorem, kink-antikink collisions, radiation.

NLS: 2.4.2, 7.4 and 7.5.

Lecture 25

PERTURBATION ANALYSIS OF FITZHUGH-NAGUMO IMPULSES:
Studies of the stable and unstable solutions, bifurcation points.

NLS: 7.6.1.

Lecture 26

COUPLED NERVE IMPULSES:
Ephaptic interactions, synchronization of impulse speeds.

NLS: 7.6.2.

Lecture 27

EMERGENT STRUCTURES IN BIOLOGY:
The nature of life, hierarchical structures, immense numbers, nonlinearity and coherent structures.

NLS: Chapter 9.

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