WISM536 : Seminar on KdV

Heinz Hanßmann, Johan van de Leur, Paul Zegeling

fall	time	place
lectures	friday 11:00 - 13:00	HFG 611

ECTS : 7.5 credit points

No seminar on fridays 27 september and 6 december. Until 1 november the seminar takes place in BBL 069, with the exceptions of friday 11 october when the seminar takes place in BBL 071 and of friday 25 october when the seminar is shifted by 1 hour to 12:00 - 14:00 and takes place in HFG 610. From friday 8 november on the seminar takes place in HFG 611.

The Korteweg de Vries (KdV) equation describes waves on shallow water surfaces. It is particularly notable as a non-linear partial differential equation that nonetheless can be exactly solved; the solutions include solitons. The mathematical theory behind the KdV equation is rich and interesting, and a topic of active mathematical research. The seminar aims to bring together several aspects of the KdV equation, notably algebraic, analytic and numerical aspects.


Each week one lecture is given on a particular topic. The lecturer also constructs an exercise for all other students, which is not too difficult (at least, not more than one or two hours work). Students have to hand in these exercises one week later, and who constructed the exercise grades the exercise on a scale from 0-10.

Assumed knowledge

A good basic knowledge of
algebra/group theory
differential equations/dynamical systems
numerical analysis
(at least one of these, but preferably two or all three)

Examination

The presentations (80%) and the home work excercises (20%).

Subjects for presentation

1. "Dynamical systems theory for finding travelling wave solutions": Johan van de Leur
2. "Lax equation and KdV hierarchy": Jan Boogert
3. "KP hierarchy and reduction to KdV": Jan Boogert
4. "Hirota equations and tau functions":
5. "Dirac sea and KP/KdV hierarchy":
6. "Poisson structure and invariant tori": Felix Beckebanze
7. "Action angle variables and Birkhoff co-ordinates": Heinz Hanßmann
8. "Perturbations of the KdV equation": Felix Beckebanze
9. "Infinite-dimensional KAM theory":
10. "Unexpected instability": Felix Beckebanze
11. "Semi-analytic methods for solitary and travelling waves": Joost van Dijk
12. "Symplectic and multisymplectic schemes for the KdV equation": Jan Boogert
13. "Spectral methods, stability and time-stepping schemes": Jan Boogert
14. You may have your own suggestion: please discuss with us

Literature

U.M. Ascher

Numerical methods for evolutionary differential equations

SIAM (2008)

L.A. Dickey

Soliton Equations and Hamiltonian Systems

World scientific (1991)

P.G. Drazin and R.S. Johnson

Solitons: an introduction

Cambridge texts in applied mathematics (1989)

E. M. de Jager

On the origin of the Korteweg-de Vries equation

Forum der Berliner Mathematischen Gesellschaft 19, p. 171-195 (2011)

V G. Kac and A.K. Raina

Bombay Lectures on Highest Weight Representations

of Infinite Dimensional Lie Algebras

World Scientific, vol. 2 of Adv. Ser. in Math. Phys. (1987)

T. Kappeler and J. Pöschel

KdV & KAM

Springer (2003)

T. Miwa, M. Jimbo and E. Date

Solitons

Cambridge University Press (2000)

M. Remoissenet

Waves called solitons

Springer (1996)

L. Trefethen

Spectral methods in Matlab

SIAM (2000)

Contents

13. September. Introduction, distribution of (remaining) talks.

20. September. Dynamical systems theory for finding travelling wave solutions.

4. October. Poisson structure and invariant tori.

11. October. Action angle variables and Birkhoff co-ordinates.

18. October. Lax equation and KdV hierarchy.

25. October. Unexpected instability.

1. November. Semi-analytic methods for solitary and travelling waves.

8. November. Symplectic and multisymplectic schemes for the KdV equation.

15. November. Perturbations of the KdV equation.

22. November. KP hierarchy and reduction to KdV.

29. November. Cancelled.

13. December. Spectral methods, stability and time-stepping schemes.