A Simple Example: Plane Wave Solution:

1
Stability of Vortex Filaments in the Localized
Induction Approximation
By Scotty Keith, Annalisa Calini, and Stephane
Lafortune
Linearized VFE Using the squared eigenfunctions
of the AKNS system and the relation between the
VFE and NLS equations, we construct solutions of
the linearized VFE equation  and relate the
stability properties of vortex filaments to those
of the associated NLS potentials. We illustrate
this approach in the case of multiply-covered
circles, corresponding to plane wave potentials
the simplest spatially independent solutions of
the NLS equation. We find that the solution of
the linearized VFE where for a
solution to the AKNS and ,
, for
A Simple Example Plane Wave Solution One of the
simplest solutions to the NLS is the plane wave
solution The linearized NLS around the plane
wave solution is Its solutions can be obtained
through the squared eigenfunctions with
special cases at and for
as This family of solutions can
alternatively be obtained using a Fourier series
expansion.
Introduction The NLS equation is a fundamental
example of soliton equation, with solitary wave
solutions constructed using inverse scattering
techniques. The NLS can be written as q(x,t),
the complex potential, is a function of space x
and time t.
At the heart of the inverse scattering method for
the NLS equation is the AKNS system an
eigenvalue problem and an evolution equation for
a vector-valued eigenfunction, whose solvability
condition is the NLS equation itself. The AKNS
system is For a vector valued
eigenfunction, and the eigenvalue
parameter. The condition is equivalent to NLS.
(b)
(a)
The Vortex Filament Equation In its simplest
form, the self-induced dynamics of a vortex
filament in a perfect fluid is governed by the
Vortex Filament Equation (VFE), which can be
written as where for or
as where The VFE  is related to the cubic,
focusing Nonlinear Schrödinger (NLS) equation via
a change of variables known as the Hasimoto map.
The inverse of the Hasimoto map translates a
solution of the NLS into a solution to the
VFE where is a 2X2 matrix whose columns
solve the AKNS system.
A solution of the NLS equation is said to be
stable if any solution which is close to at
time t0 remains close to for all
times.  Linear Stability In this work, we study
the linear stability" of solutions of the NLS
equation, defined as follows.  Assume
where is assumed to remain small for some
finite time. Substituting q(x,t) into the NLS
equation, and removing the higher order terms of
, we obtain the Linearized NLS equation If
we can solve the LNLS or understand the growth
rate of its solutions in time, we can then
formulate conclusions about the linear stability
of the base solution .
(c)
Plane Wave solution for the VFE The solution the
VFE that corresponds to the plane wave solution
to the NLS can be written as When graphed, it
is a circle that propagates in a direction
perpendicular the plane in which it inhabits, at
a speed that is inversely proportional to its
radius.
Squared Eigenfunctions The squared
eigenfunctions of the AKNS system play a central
role in linear stability studies of solutions of
the NLS equation, as they provide a large set of
solutions of the linearization of the NLS
equation about a given solution. Let and
both solve the AKNS. The squared
eigenfunctions are defined as , ,
and . We can show that solves
the linearized NLS.

• Figure
• Perturbations of a circle moving upward in time.
• A stable perturbation of a simply covered
  circle.
• A stable perturbation of a simply covered
  circle.
• An unstable perturbation of a multiply covered
  circle, growing exponentially in time.

t
t