Geodesic
Nonlocal reductions of the~Ablowitz--Ladik equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 197 (2018) no. 1, pp. 24-44

Voir la notice de l'article provenant de la source Math-Net.Ru

Our purpose is to develop the inverse scattering transform for the nonlocal semidiscrete nonlinear Schrödinger equation (called the Ablowitz–Ladik equation) with $\mathcal{PT}$ symmetry. This includes the eigenfunctions (Jost solutions) of the associated Lax pair, the scattering data, and the fundamental analytic solutions. In addition, we study the spectral properties of the associated discrete Lax operator. Based on the formulated (additive) Riemann–Hilbert problem, we derive the one- and two-soliton solutions for the nonlocal Ablowitz–Ladik equation. Finally, we prove the completeness relation for the associated Jost solutions. Based on this, we derive the expansion formula over the complete set of Jost solutions. This allows interpreting the inverse scattering transform as a generalized Fourier transform.
Keywords: integrable system, PT symmetry, nonlocal reduction, Riemann–Hilbert problem.
Mots-clés : soliton 
@article{TMF_2018_197_1_a1,
     author = {G. G. Grahovski and A. Mohammed and H. Susanto},
     title = {Nonlocal reductions of {the~Ablowitz--Ladik} equation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {24--44},
     publisher = {mathdoc},
     volume = {197},
     number = {1},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2018_197_1_a1/}
}
TY  - JOUR
AU  - G. G. Grahovski
AU  - A. Mohammed
AU  - H. Susanto
TI  - Nonlocal reductions of the~Ablowitz--Ladik equation
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2018
SP  - 24
EP  - 44
VL  - 197
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TMF_2018_197_1_a1/
LA  - ru
ID  - TMF_2018_197_1_a1
ER  - 
%0 Journal Article
%A G. G. Grahovski
%A A. Mohammed
%A H. Susanto
%T Nonlocal reductions of the~Ablowitz--Ladik equation
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2018
%P 24-44
%V 197
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TMF_2018_197_1_a1/
%G ru
%F TMF_2018_197_1_a1
G. G. Grahovski; A. Mohammed; H. Susanto. Nonlocal reductions of the~Ablowitz--Ladik equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 197 (2018) no. 1, pp. 24-44. http://geodesic.mathdoc.fr/item/TMF_2018_197_1_a1/