Solitons

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Numbers in italics refer to sections (§) or questions (Q) from the textbooks by Drazin and Johnson (DJ),
Manton and Sutcliffe (MS), or Dauxois and Peyrard (DP), where you can find out about the material to be covered.

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Lecture outlines

Week Monday 11am CG83 Tuesday 10am E101

1 1 Introduction
1.1 What is a soliton?
The KdV equation DJ §1.1, §1.2; DP §1.1 Basic properties of solitons DJ §1.2, §1.3
1.2 The ball and box model

2 2 Waves, dispersion and dissipation DJ §1.1
2.1 Dispersion
Examples; phase and group velocities
2.2 The Gaussian wavepacket
2.3 Dissipation DJ §1.1
2.4 Summary

3 3 Travelling waves DJ §2.1, §2.2
3.1 The KdV soliton
3.2 The sine-Gordon kink
3.3 Physical model for the sine-Gordon kink
DJ Q8.2; DP §2.1

4 4 Topological lumps and the Bogomolnyi bound
4.1 The sine-Gordon kink as a topological lump MS §5.3; DP §2.2.1
4.2 The Bogomolnyi argument MS §5.1 4.3 Summary
5 Conservation laws
Introduction

5 5.1 The basic idea DJ §5.1.1
5.2 Example: conservation of energy for sine-Gordon
5.3 Conserved quantities for KdV DJ §5.1.1
5.4 The Gardner transform DJ §5.1.2

6 The Gardner transform (concluded)
(NB: section 5.5, which you can find in the printed notes, is
non-examinable bonus material) 6 Bäcklund transformations DJ §5.4
6.1 Definition
6.2 A simple example

7 6.3 The Bäcklund transformation for sine-Gordon DJ §5.4.1
6.4 First example: the sG kink from the vacuum
6.5 The theorem of permutability

8 6.6 The two-soliton solution
6.7 Asymptotics of multisoliton solutions Asymptotics of multisoliton solutions (concluded)

9 6.8 The breather 7 The Hirota method DJ §5.3
7.2 KdV in bilinear form DJ §5.3.1
7.2.1 The quadratic form

10 7.2.2 Hirota's bilinear operator D(f·g)
7.3 Solutions
7.3.1 First example - the 1-soliton 7.3.2 The N-soliton solution (sketch)
7.4 Phase shifts for the KdV 2-soliton
The end!

Lecture notes

Material for MSc (Solitons V) students

Patrick Dorey

Last modified: 2 December 2024

Week	Monday 11am CG83	Tuesday 10am E101
1	1 Introduction 1.1 What is a soliton? The KdV equation DJ §1.1, §1.2; DP §1.1	Basic properties of solitons DJ §1.2, §1.3 1.2 The ball and box model
2	2 Waves, dispersion and dissipation DJ §1.1 2.1 Dispersion Examples; phase and group velocities	2.2 The Gaussian wavepacket 2.3 Dissipation DJ §1.1 2.4 Summary
3	3 Travelling waves DJ §2.1, §2.2 3.1 The KdV soliton 3.2 The sine-Gordon kink	3.3 Physical model for the sine-Gordon kink DJ Q8.2; DP §2.1
4	4 Topological lumps and the Bogomolnyi bound 4.1 The sine-Gordon kink as a topological lump MS §5.3; DP §2.2.1 4.2 The Bogomolnyi argument MS §5.1	4.3 Summary 5 Conservation laws Introduction
5	5.1 The basic idea DJ §5.1.1 5.2 Example: conservation of energy for sine-Gordon 5.3 Conserved quantities for KdV DJ §5.1.1	5.4 The Gardner transform DJ §5.1.2
6	The Gardner transform (concluded) (NB: section 5.5, which you can find in the printed notes, is non-examinable bonus material)	6 Bäcklund transformations DJ §5.4 6.1 Definition 6.2 A simple example
7	6.3 The Bäcklund transformation for sine-Gordon DJ §5.4.1 6.4 First example: the sG kink from the vacuum	6.5 The theorem of permutability
8	6.6 The two-soliton solution 6.7 Asymptotics of multisoliton solutions	Asymptotics of multisoliton solutions (concluded)
9	6.8 The breather	7 The Hirota method DJ §5.3 7.2 KdV in bilinear form DJ §5.3.1 7.2.1 The quadratic form
10	7.2.2 Hirota's bilinear operator D(f·g) 7.3 Solutions 7.3.1 First example - the 1-soliton	7.3.2 The N-soliton solution (sketch) 7.4 Phase shifts for the KdV 2-soliton The end!