Title: Interplay between the Inverse Scattering Method and Fokas's Unified Transform with an Application,
Abstract: It is known that the initial-boundary value problem for certain integrable
partial differential equations (PDEs) on the half-line with integrable boundary
conditions can be mapped to a special case of the Inverse Scattering Method
(ISM) on the full-line. This can also be established within the so-called
Unified Transform (UT) for initial-boundary value problems with linearizable
boundary conditions. In this paper, we show a converse to this statement within
the Ablowitz-Kaup-Newell-Segur (AKNS) scheme: the ISM on the full-line can be
mapped to an initial-boundary value problem with linearizable boundary
conditions. To achieve this, we need a matrix version of the UT that was
introduced by the author to study integrable PDEs on star-graphs. As an
application of the result, we show that the new, nonlocal reduction of the AKNS
scheme introduced by Ablowitz and Musslimani to obtain the nonlocal Nonlinear
Schrödinger (NLS) equation can be recast as an old, local reduction, thus
putting the nonlocal NLS and the NLS equations on equal footing from the point
of view of the reduction group theory of Mikhailov.