Vector Nonlinear Schr\"odinger Equation on the half-line

TL;DR

This paper studies the vector nonlinear Schrödinger equation on a half-line, deriving integrable boundary conditions, constructing the inverse scattering method, and revealing unique vector soliton interactions with boundaries.

Contribution

It introduces new integrable boundary conditions for the vector nonlinear Schrödinger equation and develops an inverse scattering framework incorporating boundary effects.

Findings

Derived two classes of integrable boundary conditions.

Constructed the inverse scattering method with boundary conditions.

Discovered vector-specific soliton boundary interactions.

Abstract

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.