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In this paper, we develop an inverse scattering transform for the integrablefocusing nonlinear Schr\"odinger (NLS) equation on the half-line subject to aclass of boundary conditions. The method is based on the notions of integrableboundary conditions and double-row monodromy matrix, developed by Sklyanin,which characterize initial-boundary value problems for NLS on an interval. Itfollows from Sklyanin's approach that a hierarchy of integrable boundaryconditions can be derived based on a semi-discrete type Lax pair involving thetime-part Lax matrix of NLS and a reflection matrix. The inverse scatteringtransform relies on the formulation of a scattering system for the double-rowmonodromy matrix characterizing an interval problem. The scattering system forthe half-line problem for NLS is obtained by extending one endpoint of theinterval to infinity. Then, we derive spectral and analytic properties of thescattering systems, and set up the inverse part using a Riemann-Hilbertformulation. We also show that our approach is equivalent to a nonlinear methodof reflection by extending the initial-boundary value problem on the half-lineto an initial value on the whole axis. Explicit examples of soliton solutionson the half-line are provided. Although we only consider the NLS model as aparticular example, the method we present in this paper can be readily appliedto a wide range of integrable PDEs on the half-line.

On an inverse scattering transform for the nonlinear Sch\"odinger equation on the half-line