Vector Nonlinear Schrödinger Equation on the half-line

Abstract

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.

23 pages, 5 figures, some clarifications in propositions 3.1 and 3.2, added appendix with detailed comparison between linear and nonlinear cases. Accepted in J. Phys. A

References in corpus (4)

• N-soliton collision in the Manakov model
• On a systematic approach to defects in classical integrable field theories
• Solitons, boundary value problems and a nonlinear method of images
• Spontaneous symmetry breaking in the gl(N)-NLS hierarchy on the half line

Cited by in corpus (7)

• Set-theoretical reflection equation: Classification of reflection maps
• From Hamiltonian to zero curvature formulation for classical integrable boundary conditions
• New integrable boundary conditions for the Ablowitz-Ladik model: from Hamiltonian formalism to nonlinear mirror image method
• Spectrally accurate space-time solution of Manakov systems
• Nonlinear mirror image method for nonlinear Schrödinger equation: Absorption/emission of one soliton by a boundary
• Classical $N$-Reflection Equation and Gaudin Models
• Nonlinear Schrödinger equation on the half-line without a conserved number of solitons