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 Kortewegde 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 splitstep 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
ADDITIONAL REFERENCES:
5 P.L. Sulem and C. Sulem, ``Nonlinear Schrodinger Equation:
SelfFocusing and Wave Collapse," Springer Verlag, 1999.
6 A. Fordy (editor), ``Soliton theory: a survey of results,"
Manchester
University Press, 1990. (good chapter on tests of complete
integrability)
7 A. Newell and J.V. Moloney, ``Nonlinear Optics," AdisonWesley, 1992.
8 T. Dauxois and M. Peyrard, ``Physics of Solitons," Cambridge Univ.
Press, 2006.
INTERNET POPULAR READING ON SOLITARY WAVES:
SCHEDULE OF TALKS:
week
nr.  date

topic

speaker

2  31.10.2006 
introduction (PDF)

T. Dohnal

3  7.11.2006

phase velocity,
group velocity, smoothing via dispersion (PDF
 scanned)

T. Dohnal 
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)

T. Dohnal 
5  21.11.2006 
SplitStep Methods (PDF)

M. Koller

6  28.11.2006 
SineGordon Equation (PDF)

D. Gablinger

7  5.12.2006 
Cubic NLS and CubicQuintic NLS (PDF)

M. Sani

8  12.12.2006

Kortewegde Vries Equation (PDF)

C. Gittelson

9  19.12.2006 
Hamiltonian structure of ODEs
and PDEs (PDF)

T. Dohnal

11  9.1.2007 
Shallow Water Wave Solitons 
tidal bores, tsunamis  CANCELLED

M. Cada

12  16.1.2007

Modulational Instability and
Freak Waves (PDF)

C.Varsakelis

13  23.1.2007

Analysis of the Petviashvili
iteration method (handout
PDF, additional
calculations, overview)

P. Kauf

14  30.1.2007

Collapse Phenomenon in NLS;
Townes soliton (PDF)

J. Kowalski
