# winter semester 2006/07

Day, Time, Place: Tuesdays 17:00 - 18:45  HG G 26.3       (the start time is 17:00 sharp)
Start Date: 31.10.2006
Instructor: Tomas Dohnal
Contact:
dohnal -at- math.ethz.ch

DESCRIPTION:
Solitary waves are special solutions to nonlinear PDEs which arise due to a perfect balance between linear dispersive and nonlinear effects. They are localized disturbances that, as the name suggests, evolve without any change to their shape. In cases of completely integrable PDEs they are called solitons. Solitary waves appear in real world as, for instance, laser generated pulses, tidal bores, morning glory clouds, freak waves, tsunami, wakes of high speed ships, etc.

In this seminar, after briefly covering the history of solitary wave research, we will define a plane wave, phase velocity, wavepacket, group velocity, dispersion relation and the slowly varying envelope approximation. We will next derive some famous soliton carrying PDEs, like the Korteweg-de Vries and the Nonlinear Schroedinger equations and sutdy their Hamiltonian structure, the simplest explicitely known solitons, Backlund transformations, hierarchy of conserved quantities, etc.

We will also concentrate on numerical methods for finding solitary wave solutions in cases when analytic methods fail or are too complicated. The methods include Newton iteration, fixed point iterations, the reduced variational principle and relaxation methods. Another topic in numerics will be the use of split-step and pseudospectral methods for time evolution of the governing PDEs.

FORMAT OF THE SEMINAR:
After a couple of introductory lectures by myself each student will a 60'-70' presentation on a selected topic. Most projects will include a Matlab programming task.

SEMINAR MOST USEFUL FOR SOMEONE WHO HAS:
Basic knowledge of PDEs and some experience with Matlab (or good knowledge of some lower level programming language). Some experience with numerical methods for PDEs and ODEs is a plus.

PRIMARY REFERENCES:
1- P.G. Drazin and R.S. Johnson, "Solitons: an introduction," (Cambridge Univ. Press, 1989).
2- G.B. Whitham, "Linear and Nonlinear Waves" (Wiley, New York, 1974).
3- A.C. Scott, "Nonlinear Science: Emergence and Dynamics of Coherent Structures," 2nd ed., Oxford University Press, Oxford, 2003.
4- various scientific articles and internet sources