https://doi.org/10.1140/epjb/e2002-00284-8
New geometries associated with the nonlinear Schrödinger equation
The Institute of Mathematical Sciences, Chennai 600 113, India
Corresponding author: a radha@imsc.ernet.in
Received:
5
December
2001
Published online:
2
October
2002
We apply our recent formalism
establishing new connections
between the geometry of moving space curves and soliton equations,
to the nonlinear
Schrödinger equation (NLS).
We show that any given solution of the NLS gets associated with
three distinct space curve evolutions. The tangent
vector of the first of these curves, the binormal vector of the second
and the normal vector of the third, are shown to
satisfy the integrable Landau-Lifshitz (LL) equation
, (
).
These connections enable us to find the three surfaces
swept out by the moving curves associated with
the NLS. As an example, surfaces corresponding to a
stationary envelope soliton solution of the NLS are obtained.
PACS: 02.40.Hw – Classical differential geometry / 05.45.Yv – Solitons / 75.10.Hk – Classical spin models
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2002
