Solitons

This is the web-page for the 2024-25 level III Solitons course in the Michaelmas term. See also the ultra page for the course.

Contact

• Patrick Dorey
• Room No: MCS 2033
• email: p.e.dorey@durham.ac.uk

Lectures

• Monday 11:00 in CG83 and Tuesday 10:00 in E101
• Lecture outlines and printed notes

Office hour

• Tuesday 12:00 in MCS2033
  (If you have a question and can't make it then, please send me an email and we can fix up an alternative.)

Problems classes

• Thursday 9:00 in MCS2068 in teaching weeks 4, 6, 8 and 10
  • Week 4: problems 10, 13(e), 13(b)
  • Week 6: problem 23
  • Week 8: problem 27

Animations and computer programs

• Click here to see numerical solutions of the "dispersing KdV" equation.
• Click here to see numerical solutions of the dispersionless KdV, or Monge, equation.
• Click here to see numerical solutions of the full KdV equation.
• Click here to see the other animations, of solutions of the full KdV equation, that will be discussed in the first lecture.
  (Warning - these may slow down your computer while they load!)
• An animation of a wavepacket.
• Another wavepacket, showing the effect of dispersion.
• A python program to solve the KdV equation numerically.
• Click here for animations of a kink-antikink solution to the sine-Gordon equation, and here for its energy density;
  click here for a kink-kink solution, and here for its energy density.
• Click here for contour plots of two-soliton scatterings and breathers.
• Animations of the field and energy density of a breather solution to the sine-Gordon equation are here.
• An animation of a two-wave solution of Burgers' equation.

Problems

• Problem set 1 (problems 1-13)
• Problem set 2 (problems 14-25)
• Problem set 3 (problems 26-44)
• Useful integrals

Assignments

Assignments will be set every week, but only the even-numbered ones will be for handing in. Solutions to all will be posted on Ultra as term goes by.

• Assignment 1: problems 1 and 2
• Assignment 2: problems 4 and 11, to be submitted on gradescope by 12 noon on Monday 21/10/24
• Assignment 3: problems 10 and 13(a)
• Assignment 4: problem 13(c), to be submitted on gradescope by 12 noon on Monday 4/11/24
• Assignment 5: problem 16
• Assignment 6: problem 21, to be submitted on gradescope by 12 noon on Monday 18/11/24
• Assignment 7: problem 25
• Assignment 8: problem 26, to be submitted on gradescope by 12 noon on Monday 2/12/24

Recommended books

• P.G. Drazin and R.S. Johnson, Solitons: An Introduction, CUP 1989; ISBN 0521336554
• N. Manton and P. Sutcliffe, Topological Solitons, CUP 2004; ISBN 0521838363
• T. Dauxois and M. Peyrard, Physics of Solitons, CUP 2006; ISBN 0521854210

Interesting links

• John Scott Russell
  A page about the discovery of the soliton
• Real solitons in the Strait of Gibraltar and on the Dordogne river
  
• Solitons in space
  A slightly less realistic -and much less accurately-described - example of a soliton (from this episode of the Star Trek TV programme)
• Solitons in South Africa
  A video made with students from the African Institute for Mathematical Sciences
• The Solitons Homepage
  A large set of links, with some animations
• Solitons and Soliton Collisions
  Covers sine-Gordon collisions, with very nice animations of the pendulum model to be discussed in lectures
• Making waves - singly
  An article about the applications of solitons
• Mathematics at the turn of the millennium
  A general article about modern mathematics with some interesting comments about soliton theory about half-way down
• The Fermi-Pasta-Ulam-Tsingou problem
  This was one of the reasons for the revival of interest in solitons in the 1960s. You can read some more about it here
• Tsunamis
  An article about the equations which describe Tsunamis (including some comments on the links - or not - with solitons)
• Tony Skyrme and the Origins of Skyrmions
  An interesting historical article from 2020 about Skyrme's work on solitons
• The 2006 Steele Prizes
  In 2006 the Seminal Contribution to Research prize was awarded to Gardner, Greene, Kruskal and Miura for their work on the KdV equation

Dramatis personae

Click on the mugshots to find out more...