DEPARTMENT OF MATHEMATICAL SCIENCES

APPLIED AND COMPUTATIONAL MATHEMATICS SEMINAR

**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