Fall 2009 Reading Group on the Evans Function
Meeting time:: Mondays 2:303:30pm in MCS 180
Organizers: Margaret
Beck (mabeck at math.bu.edu) and Aaron Hoffman (ah1 at math.bu.edu)
The focus of this reading group will be the Evans function, including
theory and applications in a variety of settings. Weekly topics and
associated reading materials will be posted here.
 Week 1: Monday Sept 21 Aaron gave an introductory
lecture on what the Evans function is and how it fits into the
framework of more general stability theory. The example he worked
through is here:
 A basic example of an explicit Evans function, found here
on Björn Sandstede's webpage.
If you would like to see more explicit examples, see
Related references for your further information include:
 "Stability of
travelling waves," by Björn Sandstede. This is a good review
article and general
reference for stability theory. The [.pdf]
file can be found on Björn's webpage.
 Week 2: Monday Sept 22 Margaret will continue our
introduction to the Evans function by giving a more formal definition of the Evans function
and an overview of the historical developement of the Evans
function, including recent research topics regarding it. The
definition she will present is essentially that found in
You can download a reference
list of the papers that will be mentioned today, including
addition references to papers from the last thirty or so years that have
used the Evans function. Note: this list is not meant to be complete!
 Week 3: Monday, Oct 5 Margaret will begin to present a rigorous
construction of the Evans function using exponential dichotomies.
 Week 4: Tuesday, Oct 13 (Monday schedule) Margaret will
continue the construction of the Evans function via exponential dichotomies.
 Week 5: Monday, Oct 19 Margaret will
finish the construction of the Evans function via exponential dichotomies.
 Week 6: Monday, Oct 26 Aaron will present results from
Pego and Weinstein's 1992 paper "Eigenvalues, and instabilities of solitary
waves," which you can find here.
He will focus primarily on sections 2b (pp 7174), on the generalized KdV equation,
and 4 (pp 8388), on transitions to instability. As preparation he will also discuss those
parts of section 1 (pp 5469) which we have not already covered, in particular section 1g (pp 6568).
Sections zero (pp 4853) and 2a (pp 7071) outline the philosophy and
strategy, and it could be helpful if you read these before our
meeting.
 Week 7: Monday, Nov 2 Aaron will continue discussing the
paper by Pego and Weinstein.
 Week 8: Monday, Nov 9 Margaret will present the Gap
Lemma, which tells one how to analytically extend the Evans function
into the essential spectrum. The original papers for this are by
Kapitula and Sandstede and Gardner and Zumbrun, but she will follow
the presentation in section 4.3 of the review paper by Sandstede
(linked above).
 Week 9: Monday, Nov 16 Margaret will present one of the
examples in the paper by Kapitula and Sandstede, in which
perturbations of NLS are analyzed using extensions of the Evans
function.
 Week 10: Monday, Nov 23 Nick will speak on the paper
"Spectral stability analysis for periodic traveling wave solutions of
NLS and CGL perturbations," by T. Ivey and S. Lafortune, which you can find
here.
 Week 11: Monday, Nov 30 Nick will continue his
presentation of the paper
"Spectral stability analysis for periodic traveling wave solutions of
NLS and CGL perturbations."
