# Interplay between the Inverse Scattering Method and Fokas's Unified Transform with an Application

## TL;DR

This paper explores the relationship between the Inverse Scattering Method and Fokas's Unified Transform, demonstrating how the full-line ISM can be related to boundary value problems, with applications to nonlocal and local NLS equations.

## Contribution

It establishes a converse mapping between the full-line ISM and boundary value problems within the AKNS scheme using a matrix version of the UT.

## Key findings

- Recasts nonlocal NLS as a local reduction.
- Shows the ISM on the full-line can be mapped to boundary value problems.
- Bridges the gap between nonlocal and local integrable equations.

## 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 Schr\"odinger (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.