Fall 2009 Reading Group on the Evans Function Meeting time:: Mondays 2:30-3:30pm 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 The beginning of section 4 of "Stability analysis of pulses via the Evans function: dissipative systems," by Todd Kapitula. The [.pdf] can be found on Todd's webpage. In addition to containing an explicit example, this is a good review article focused on the Evans function. Here is another basic example: spectral stability of the traveling wave (also know as a viscous shock in this case) in Burgers equation. [.pdf] 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 "Stability of travelling waves," by Björn Sandstede, section 4.1. [.pdf] 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 71-74), on the generalized KdV equation, and 4 (pp 83-88), on transitions to instability. As preparation he will also discuss those parts of section 1 (pp 54-69) which we have not already covered, in particular section 1g (pp 65-68). Sections zero (pp 48-53) and 2a (pp 70-71) 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."