Vector nonlinear Schrödinger equation on the half-line

We investigate the Manakov model or, more generally, the vector nonlinear Schrödinger equation on the half-line. Using a Bäcklund transformation method, two classes of integrable boundary conditions are derived: mixed Neumann/Dirichlet and Robin boundary conditions. Integrability is shown by constru...

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Bibliographic Details
Published in:Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2012-03, Vol.45 (10), p.105201-21
Main Authors: Caudrelier, V, Zhang, Q C
Format: Article
Language:English
Subjects:
Boundaries
Boundary conditions
Construction
Dirichlet problem
Exact sciences and technology
integrable boundary conditions
inverse scattering method
Mathematical analysis
Mathematical models
nonlinear Schrödinger equation
Nonlinearity
Physics
Schroedinger equation
solitons
Vectors (mathematics)
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Summary:We investigate the Manakov model or, more generally, the vector nonlinear Schrödinger equation on the half-line. Using a Bäcklund 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.
ISSN:1751-8113
1751-8121
DOI:10.1088/1751-8113/45/10/105201