Speaker:Annalisa Calini, College of Charleston and the National Science Foundation
Title: Integrable dynamics of closed vortex filaments: finite-gap solutions and their linear stability

Abstract: The vortex filament equation (VFE), describing the self-induced motion of a vortex filament in an ideal fluid, is a simple but important example of integrable curve dynamics, and one in which knotted curves arise as solutions of a differential equation possessing a rich geometrical structure. The connection between the VFE and the cubic focusing nonlinear Schrodinger (NLS) equation through the well-known Hasimoto map allows the use of many of the tools of soliton theory to study properties of its solutions. In this talk I will discuss closed finite-gap filaments and their linear stability. In particular, I will use isoperiodic deformations of spectral data to construct families of periodic small amplitude finite-gap solutions of increasingly high genus, and provide a complete description of their knot types in terms of spectral data. I will then describe the relation between the linearizations of the VFE and the NLS equation, and the role of squared eigenfunctions in determining the stability properties of VFE solutions in terms of those of the associated NLS potentials. This work is part of on-going collaborations with Tom Ivey and Stephane Lafortune, both at the College of Charleston.

Time: Friday, Dec. 2, 2011, 1:30-2:30 p.m.

Place: Science and Tech I, Room 242

Department of Mathematical Sciences
George Mason University
4400 University Drive, MS 3F2
Fairfax, VA 22030-4444
http://math.gmu.edu/
Tel. 703-993-1460, Fax. 703-993-1491