The Ablowitz–Ladik system with linearizable boundary conditions *This article is dedicated to Mark Ablowitz on the occasion of his seventieth birthday. (19th August 2015)

• Record Type: Journal Article
• Authors: Biondini, Gino; Bui, Anh
• Is Part Of: Journal of physics. Volume 48:Number 37(2015)
• Publication Date: 2015-08-19
• Subjects: solitons -- inverse scattering transform -- boundary value problems -- linearizable boundary conditions -- Backlund transformations; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8113/48/37/375202
• Languages: English
• ISSNs: 1751-8113

Abstract

Abstract: The boundary value problem (BVP) for the Ablowitz–Ladik (AL) system on the natural numbers with linearizable boundary conditions is studied. In particular: (i) a discrete analogue is derived of the Bäcklund transformation that was used to solved similar BVPs for the nonlinear Schrödinger equation; (ii) an explicit proof is given that the Bäcklund-transformed solution of AL remains within the class of solutions that can be studied by the inverse scattering transform; (iii) an explicit linearizing transformation for the Bäcklund transformation is provided; (iv) explicit relations are obtained among the norming constants associated with symmetric eigenvalues; (v) conditions for the existence of self-symmetric eigenvalues are obtained. The results are illustrated by several exact soliton solutions, which describe the soliton reflection at the boundary with or without the presence of self-symmetric eigenvalues.