Friday, February 3, 2012

1110.2990 (V. Caudrelier et al.)

Vector Nonlinear Schrödinger Equation on the half-line    [PDF]

V. Caudrelier, Q. C. Zhang
We investigate the Manakov model or, more generally, the vector nonlinear
Schr\"odinger equation on the half-line. Using a B\"acklund transformation
method, two classes of integrable boundary conditions are derived: mixed
Neumann/Dirichlet and Robin boundary conditions. Integrability is shown by
constructing a generating function for the conserved quantities. We apply a
nonlinear mirror image technique to construct the inverse scattering method
with these boundary conditions. The important feature in the reconstruction
formula for the fields is the symmetry property of the scattering data emerging
from the presence of the boundary. Particular attention is paid to the discrete
spectrum. An interesting phenomenon of transmission between the components of a
vector soliton interacting with the boundary is demonstrated. This is specific
to the vector nature of the model and is absent in the scalar case. For
one-soliton solutions, we show that the boundary can be used to make certain
components of the incoming soliton vanishingly small. This is reminiscent of
the phenomenon of light polarization by reflection.
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