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,
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.
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
5- P.L. Sulem and C. Sulem, ``Nonlinear Schrodinger Equation:
Self-Focusing and Wave Collapse," Springer Verlag, 1999.
6- A. Fordy (editor), ``Soliton theory: a survey of results,"
University Press, 1990. (good chapter on tests of complete
7- A. Newell and J.V. Moloney, ``Nonlinear Optics," Adison-Wesley, 1992.
8- T. Dauxois and M. Peyrard, ``Physics of Solitons," Cambridge Univ.
INTERNET POPULAR READING ON SOLITARY WAVES:
SCHEDULE OF TALKS:
nr. - date
|2 - 31.10.2006
|3 - 7.11.2006
group velocity, smoothing via dispersion (PDF
|4 - 14.11.2006
|multiple scales expansion
method, derivation of the Nonlin.
Schroedinger Equation for pulses in optical fibers (PDF)
numerical iterative methods for
solitary wave computations (PDF)
|5 - 21.11.2006
||Split-Step Methods (PDF)
|6 - 28.11.2006
||Sine-Gordon Equation (PDF)
|7 - 5.12.2006
||Cubic NLS and Cubic-Quintic NLS (PDF)
|8 - 12.12.2006
|Korteweg-de Vries Equation (PDF)
|9 - 19.12.2006
||Hamiltonian structure of ODEs
and PDEs (PDF)
|11 - 9.1.2007
||Shallow Water Wave Solitons -
tidal bores, tsunamis - CANCELLED
|12 - 16.1.2007
|Modulational Instability and
Freak Waves (PDF)
|13 - 23.1.2007
|Analysis of the Petviashvili
iteration method (handout-
|14 - 30.1.2007
|Collapse Phenomenon in NLS;
Townes soliton (PDF)