Nonlinear Waves

Introduction

The Fractals & Chaos course mostly explores nonlinear systems that show either complicated spatial patterns (fractals) or complicated patterns in time (chaos). The Nonlinear Waves course looks at nonlinear systems that have both time and space dependence at the same time, and so are generally described by partial differential equations. In particular, we consider systems that show ordered structures - namely, some kind of waves. The 'waves' we study don't have to look like the waves at the beach. The spots on a cheetah's skin can also be regarded waves.

So why are they interesting to study? There are many systems in nature that are self-organizing. This means that they are made up of many small parts that interact with each other, and as a result of this interaction the system as a whole shows some kind of organized behaviour or pattern. For example, the heart is made up of millions of cells. These cells interact in some way to make the whole heart beat rhythmically and so keep us alive. How this happens and what to do when things go wrong is not fully understood, but is evidently important if we are to find a method to treat heart attacks. Another example is in the emerging field of nanotechnology. It is the dream of nanotechnologists to have miniature machines that could disassemble one object and rebuild anything we want such as a cable reaching into space that we could use to visit a space hotel cheaply and easily. In principle, millions of these miniature machines could make pollution, famine, disease, and old age things of the past. There are two obstacles to achieving this. The first is to build the machines. The second is to understand how to make the machines interact in a coordinated way to do what we want. This second problem is basically that of understanding a complex self-organizing system. The study of self-organizing systems is a very new field. Nonlinear waves are the simplest type of self-organizing system. If we can understand them, that will be one step on the way to understanding more complicated systems.

The first part of the course deals with a variety of important mathematical methods for this type of theoretical physics. In the later part, these techniques are applied to a number of physical, chemical, and biological systems that show some type of self-organization.

Prerequisites

None - but it would be useful if you had attended the Fractals & Chaos course. It is assumed that you are familiar with the basic mathematics taught to undergraduate scientists (in particular, eigenvalues, eigenvectors, integration, differential equations).

Brief syllabus

Basic techniques for determining the nature of solutions to ordinary and partial differential equations, singular and multiple-scale perturbation analysis, stability analysis, geometrical methods, elliptic functions, reaction-diffusion equations, excitable systems, pattern formation, solitons.

Objectives

By the end of the course students should be able to:

• understand the concepts listed in the detailed syllabus
• minimize the number of parameters in an equation by rescaling of variables
• find and/or determine the nature of and stability of wave solutions to nonlinear partial differential equations (PDEs)
• find approximate solutions to equations using singular and multiple-scale perturbation methods
• find symmetries, conserved quantities, and dispersion relations associated with time-dependent PDEs
• solve problems involving elliptic integrals and elliptic functions

Lectures

Week 1

Basic techniques for PDEs including finding the form of a wave, change of variables, reduced variables, tanh method.

Week 2

Singular perturbation theory.

Week 3

Multiple-scale perturbation theory.

Week 4

Phase-plane analysis for wave equations.

Week 5

Stability analysis.

Week 6

Conservation laws; shock waves; Burgers equation; Cole-Hopf transformation.

Week 7

Dispersion relations; geometrical methods; eikonal equation.

Week 8

Exam 1 (on weeks 1-7).

Week 9

Reaction-diffusion equations.

Week 10

Excitable systems.

Week 11

Pattern formation.

Week 12

Elliptic integrals and elliptic functions.

Week 13

Solitons and the KdV equation.

Week 14

Inverse scattering method and Bäcklund transformations.

Week 15

Nonlinear Schrödinger equation.

Week 16

Sine-Gordon equation.

Week 17

Exam 2 (on weeks 1-16).

Teaching methods

Lectures/tutorials and detailed printed lecture notes will be provided.

Course assessment

Homework: 4%

Exam 1: 48%

Exam 2: 48%

References

A. Scott, "Nonlinear Science: Emergence and Dynamics of Coherent Structures", OUP, 1999.

R. Knobel, "An Introduction to the Mathematical Theory of Waves", American Mathematical Society, 1999.

S.S. Shen, "A Course on Nonlinear Waves", Kluwer Academic, 1993.

P. Grindrod, "Patterns and Waves: The Theory and Applications of Reaction-Diffusion Equations", Clarendon Press, 1991.

J.D. Murray, "Mathematical Biology", 3rd edn, Springer, 2002.

P.G. Drazin, "Solitons", CUP, 1983.

Eric Infeld and George Rowlands, "Nonlinear Waves, Solitons and Chaos", 2nd edn, CUP, 2000.