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.

19 pages, title modified, references updated and some typos removed. Justification of our use of linearizable boundary conditions and clearer interpretation of the meaning of the results. Author version of the article to appear in Stud. Appl. Math

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• Nonlocal Coupled HI-MKdV Systems
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• Nonlinear mirror image method for nonlinear Schrödinger equation: Absorption/emission of one soliton by a boundary
• Nonlocal Modified KdV Equations and Their Soliton Solutions
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• The Ablowitz-Ladik system on a graph
• Hirota bilinear forms of the AKNS($N$) systems
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• Linearizable boundary value problems for the elliptic sine-Gordon and the elliptic Ernst equations